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.”
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?
Since we know there will be only one lead guitarist (Graeme) and one keyboard player (Craig) we put a one in the appropriate areas.
If 16 people try out then there will be 16 people that could be a possible guitarist. That leaves 15 people left for the drumming position because 1 out of the 16 people will become their guitarist.
Now 14 people are left for the bassist. 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?***
For this we can also use the fundamental principle of counting, so we must multiply these six numbers out.
Graeme and Craig are automatically in the band and they are both boys so we will put 1 in both their areas.
If half the people that try out are boys and half girls then there are 8 boys and 8 girls. (16/2=8) So for the last (or third) boy they will have a total of 8 to choose from.
For the first girl there will be 8 girls to choose from. Now one is chosen, there will be 7 girls left that they can choose for the second girl. For the last girl they can pick one out of the 6 remaining girls.
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.