From a squad of 12 cheerleaders, 10 will assemble themselves into a 4-level pyramid at the Homecoming rally.
1.) How many different combinations of cheerleaders can be used to build the pyramid?
For this I got 12C10=66......Right or Wrong?
2.) The coach decides that Alexis must be at the top of the pyramid, and that Rachel, Jennifer, Brittany, and Nicole will form the base of the pyramid. How many different combinations of 10 cheerleaders can now be chosen to form the pyramid?
For this I got 7C5=21....Right or Wrong?
3.) Suppose Alexis has an injury and is unable to participate. The coach replaces Alexis with Rebecca. Now how many different combinations of 10 cheerleaders can the coach select?
I got 6C5=6........Right or Wrong?
1) For the first question, the number of different combinations of cheerleaders that can be used to build the pyramid can be found using the combination formula. In this case, it would be 12C10.
12C10 = 12! / (10! * (12-10)!)
= 12! / (10! * 2!)
= (12 * 11 * 10!) / (10! * 2)
= 12 * 11 / 2
= 66
So, your answer of 66 is correct.
2) For the second question, since Alexis is fixed at the top and Rachel, Jennifer, Brittany, and Nicole are forming the base, we only need to select 5 cheerleaders from the remaining 7.
7C5 = 7! / (5! * (7-5)!)
= 7! / (5! * 2!)
= (7 * 6 * 5!) / (5! * 2)
= 7 * 6 / 2
= 42 / 2
= 21
So, your answer of 21 is correct.
3) For the third question, if Alexis is replaced with Rebecca, we still need to select 5 cheerleaders from the remaining 6.
6C5 = 6! / (5! * (6-5)!)
= 6! / (5! * 1!)
= 6
So, your answer of 6 is correct.
1.) To find the number of different combinations of cheerleaders that can be used to build the pyramid, we need to calculate the combination of 12 cheerleaders taken 10 at a time. You correctly calculated this as 12C10. The formula for combinations is nCr = n! / (r!(n-r)!), where n is the total number of items to choose from and r is the number of items to be chosen. Plugging in the values, we have 12C10 = 12! / (10!(12-10)!) = 66. Therefore, your answer of 66 is correct.
2.) If Alexis must be at the top of the pyramid and Rachel, Jennifer, Brittany, and Nicole will form the base, we have 10 cheerleaders remaining to be chosen for the other levels. Since 5 cheerleaders are already fixed in their positions, we need to choose 5 cheerleaders from the remaining 7. So, the number of combinations would be 7C5 = 7! / (5!(7-5)!) = 21. Therefore, your answer of 21 is correct.
3.) If Alexis is replaced by Rebecca, we still have 10 cheerleaders to be chosen for the other levels. However, since Rebecca is now part of the 10 cheerleaders, we only need to choose 4 more. So, the number of combinations would be 6C4 = 6! / (4!(6-4)!) = 15. Therefore, your answer should be 15 instead of 6, indicating that your answer was wrong.