3. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
4. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
5. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
6. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
7. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
The number of k-permutations (ordered arrangements) taken
from n objects is:
n!
nPk = (n – k)!
8. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
The number of k-permutations (ordered arrangements) taken
from n objects is:
n!
nPk = (n – k)!
Example C. How many 2-permutations taken from {a, b, c}
are there?
9. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
The number of k-permutations (ordered arrangements) taken
from n objects is:
n!
nPk = (n – k)!
Example C. How many 2-permutations taken from {a, b, c}
are there?
n=3, k=2,
10. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
The number of k-permutations (ordered arrangements) taken
from n objects is:
n!
nPk = (n – k)!
Example C. How many 2-permutations taken from {a, b, c}
are there?
3!
n=3, k=2, 3P2 = (3 – 2)!
11. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
The number of k-permutations (ordered arrangements) taken
from n objects is:
n!
nPk = (n – k)!
Example C. How many 2-permutations taken from {a, b, c}
are there?
3! 6
n=3, k=2, 3P2 = (3 – 2)! = 1 = 6
12. Permutations and Combinations
A k-permutation is an ordered lineup of k objects.
Example A. List all the 2-permutations taken from {a, b, c}.
ab, ba, ac, ca, bc, cb
Example B. List all the 3-permutations taken from {a, b, c}.
abc, acb, bac, bca, cab, cba
The number of k-permutations (ordered arrangements) taken
from n objects is:
n!
nPk = (n – k)!
Example C. How many 2-permutations taken from {a, b, c}
are there?
3! 6
n=3, k=2, 3P2 = (3 – 2)! = 1 = 6
They are {ab, ba, ac, ca, bc, cb}.
14. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10.
15. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7,
16. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)!
17. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
18. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
= 10 x 9 x 8 x .. X 4 = 604800 possibilities.
19. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
= 10 x 9 x 8 x .. X 4 = 604800 possibilities.
A k-combination is a (unordered) collection of k objects.
20. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
= 10 x 9 x 8 x .. X 4 = 604800 possibilities.
A k-combination is a (unordered) collection of k objects.
Example E. List all the 2-combinations from the set {a, b, c}.
21. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
= 10 x 9 x 8 x .. X 4 = 604800 possibilities.
A k-combination is a (unordered) collection of k objects.
Example E. List all the 2-combinations from the set {a, b, c}.
{a, b}, {a, c}, {b, c}
22. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
= 10 x 9 x 8 x .. X 4 = 604800 possibilities.
A k-combination is a (unordered) collection of k objects.
Example E. List all the 2-combinations from the set {a, b, c}.
{a, b}, {a, c}, {b, c}
Example F. List all the 3-combination taken from {a, b, c}.
23. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
= 10 x 9 x 8 x .. X 4 = 604800 possibilities.
A k-combination is a (unordered) collection of k objects.
Example E. List all the 2-combinations from the set {a, b, c}.
{a, b}, {a, c}, {b, c}
Example F. List all the 3-combination taken from {a, b, c}.
{a, b, c}
24. Permutations and Combinations
Example D. How many different arrangements of 7 people
from a group of 10 people in a row of 7 seats are there?
There are 10 people so n = 10. We are to seat 7 of them in
order so k = 7, so there are
10P7 = 10! / (10 – 7)! =10! / 3!
= 10 x 9 x 8 x .. X 4 = 604800 possibilities.
A k-combination is a (unordered) collection of k objects.
Example E. List all the 2-combinations from the set {a, b, c}.
{a, b}, {a, c}, {b, c}
Example F. List all the 3-combination taken from {a, b, c}.
{a, b, c}
The number of k-combinations (unordered collections) taken
from n objects is:
n!
nCk =
(n – k)!k!
27. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
C2 = 3!
3
(3 – 2)!2!
28. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 =
(3 – 2)!2!
29. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
30. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
31. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
They are {a, b}, {a, c}, {b, c}.
32. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
They are {a, b}, {a, c}, {b, c}.
Example H. A Chinese take-out offers a beef dish, a chicken
dish, a vegetable dish, fried rice, and fried noodles. We may
chose any 3 of them for a 3-Combo Special. How many
different 3-Combo Specials are possible?
33. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
They are {a, b}, {a, c}, {b, c}.
Example H. A Chinese take-out offers a beef dish, a chicken
dish, a vegetable dish, fried rice, and fried noodles. We may
chose any 3 of them for a 3-Combo Special. How many
different 3-Combo Specials are possible?
n = 5, we are to take 3 of so k = 3,
34. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
They are {a, b}, {a, c}, {b, c}.
Example H. A Chinese take-out offers a beef dish, a chicken
dish, a vegetable dish, fried rice, and fried noodles. We may
chose any 3 of them for a 3-Combo Special. How many
different 3-Combo Specials are possible?
n = 5, we are to take 3 of so k = 3, hence there are
5!
5 C3 = (5 – 3)!3!
35. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
They are {a, b}, {a, c}, {b, c}.
Example H. A Chinese take-out offers a beef dish, a chicken
dish, a vegetable dish, fried rice, and fried noodles. We may
chose any 3 of them for a 3-Combo Special. How many
different 3-Combo Specials are possible?
n = 5, we are to take 3 of so k = 3, hence there are
5! 5!
5 C3 = (5 – 3)!3! = 2!3!
36. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
They are {a, b}, {a, c}, {b, c}.
Example H. A Chinese take-out offers a beef dish, a chicken
dish, a vegetable dish, fried rice, and fried noodles. We may
chose any 3 of them for a 3-Combo Special. How many
different 3-Combo Specials are possible?
n = 5, we are to take 3 of so k = 3, hence there are
5*4
5! 5!
5C3 = (5 – 3)!3! = 2!3!
37. Permutations and Combinations
Example G. How many 2-combinations from the set {a, b, c}
are there?
n = 3, k = 2, hence there are
3! 3
3C2 = =3
(3 – 2)!2!
So there are three 2-combinations.
They are {a, b}, {a, c}, {b, c}.
Example H. A Chinese take-out offers a beef dish, a chicken
dish, a vegetable dish, fried rice, and fried noodles. We may
chose any 3 of them for a 3-Combo Special. How many
different 3-Combo Specials are possible?
n = 5, we are to take 3 of so k = 3, hence there are
5*4
5! 5!
5C3 = (5 – 3)!3! = 2!3! = 10 3-Combos specials.
38. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
39. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
40. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
41. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
42. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
These are combinations so there are 13C4 possibilities.
43. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
These are combinations so there are 13C4 possibilities.
c.We are to select two men and two women for a 4-people
committee, how many are possibilities?
44. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
These are combinations so there are 13C4 possibilities.
c.We are to select two men and two women for a 4-people
committee, how many are possibilities?
There are 2-steps to make the committee,
45. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager,
a treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
These are combinations so there are 13C4 possibilities.
c.We are to select two men and two women for a 4-people
committee, how many are possibilities?
There are 2-steps to make the committee, select the
2 women, then 2 men.
46. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager, a
treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
These are combinations so there are 13C4 possibilities.
c.We are to select two men and two women for a 4-people
committee, how many are possibilities?
There are 2-steps to make the committee, select the
2 women, then 2 men. There are 8C2 ways to select 2 women.
47. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager, a
treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
These are combinations so there are 13C4 possibilities.
c.We are to select two men and two women for a 4-people
committee, how many are possibilities?
There are 2-steps to make the committee, select the
2 women, then 2 men. There are 8C2 ways to select 2 women.
There are 5C2 ways to select 2 men.
48. Permutations and Combinations
Example I. There are 5 men and 8 women.
a. We are to select a president, a vice president, a manager, a
treasurer from them. How many different possibilities are
there?
These are permutations since the order is important.
Hence there are 13P4 possibilities.
b. We are to select 4 people for a committee, how many
different 4-people committees are possible?
These are combinations so there are 13C4 possibilities.
c.We are to select two men and two women for a 4-people
committee, how many are possibilities?
There are 2-steps to make the committee, select the
2 women, then 2 men. There are 8C2 ways to select 2 women.
There are 5C2 ways to select 2 men. Hence there are
8C2 x 5C2 ways for to select the 2-men-2-women committees.
