A group of 15 of your friends wants to go to a concert. Unfortunately, you only have 10 tickets available to give to your friends. How many different groups of friends could you take to the concert?
combinations of 15 ten at a time
n = 15
r = 10
C(n,r) = n!/[ r!(n-r)! ]
= 15! /[ 10! (5!) ]
= 15*14*13*12*11 /(5*4*3*2)
= 15/15 * 14/2 * 13 * 12/4 * 11
= 7 * 13 * 3 * 11
= 3003
To find out how many different groups of friends you could take to the concert, we can use the combination formula. The combination formula is expressed as "n choose k" and is calculated using the formula:
C(n, k) = n! / (k!(n-k)!)
Where:
- n is the total number of friends (in this case, 15)
- k is the number of tickets available (in this case, 10)
- "!" represents factorial, which means multiplying a number by all the positive integers less than it down to 1.
By plugging in the values into the combination formula, we can calculate the number of different groups of friends you could take to the concert:
C(15, 10) = 15! / (10!(15-10)!)
= 15! / (10!5!)
Now, let's calculate the factorials:
15! = 15 x 14 x 13 x 12 x 11 x 10! (notice that 10! appears in both numerator and denominator, so they cancel out)
= 15 x 14 x 13 x 12 x 11
5! = 5 x 4 x 3 x 2 x 1
Therefore, the final calculation is:
C(15, 10) = (15 x 14 x 13 x 12 x 11) / (5 x 4 x 3 x 2 x 1)
= 3,003
So, you could take 3,003 different groups of friends to the concert.