Craig and Graeme decided to start a band together, with Craig playing keyboard and Graeme playing guitar. They realized they needed more members and decided to hold auditions. Sixteen people auditioned total, and they needed to choose members to fill out the rest of the band, including a drummer, guitarist, bass player, and lead singer. The document then calculates the number of possible band combinations and considers adding a requirement to only choose 3 girls if the auditioning group was half boys and half girls.

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2. One day Craig and Graeme decided to start a band. Craig is a great pianist so he decided he’d like to play the keyboard. Graeme is a very good guitar player so he decided he’d be the lead guitarist. They both thought this would be a great idea. Craig thought they were missing something in this awesome band of their’s. So Graeme said, “I know, we’re missing a band name.” Craig replied, “Let’s be the Factorials!” “Good idea, Craig!” “Thanks, but I still think we’re missing something!” “Oh! We’re missing other people!” “Oh, good one Graeme!” “What if we hold auditions for other people to be in our band?” “Well, I say that’s a swell idea.”

3. So Craig and Graeme posted a bunch of posters everywhere. Now, they were sure to get people to audition for their new band. Craig and Graeme were really excited for all of this. Aside from the keyboard player and lead guitarist they needed a percussionist (drummer), another guitarist, a bass player and a lead singer. So only four people will be chosen for their six person band. Finally the day came and a total of sixteen people showed up. How many different possible bands could be formed?

4. Fourteen people are left because 1 will be chosen for a guitarist and the other for a drummer. Now this leaves 13 people for our singer because 3 people will be chosen for other positions. As a result, we get 43 680. That means 43 680 possible bands could be formed. ***If the 16 people are half boys and half girls how many different possible bands could be formed if they were to only choose 3 girls?***

6. In the end it was a very tough decision to choose members for their band. Craig and Graeme thought really hard about this and they made their choices. As their lead singer, they chose, Vincent. For their drummer they decided to choose, Sandy. John, was chosen as their bass player. Lastly, they chose their other guitarist, Richard. The Factorials were now complete. They will go big and become the biggest band in the world with many fans. Robert will become their number one fan and Sam will be their number two fan. They will go on tour with Danny and the Cats, Bert², and the PC40’s. The End.

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