a firm has 10 sales reps. In how many ways can they be assigned to two territories, with 7 reps in one territory and 3 in the other?
To find the number of ways the sales reps can be assigned to two territories, we can use combinatorics. In this case, since there are 10 sales reps total, we need to choose 7 of them to be in one territory and the remaining 3 to be in the other territory.
The number of ways we can choose 7 sales reps out of 10 can be found by using the combination formula, which is given by:
C(n, r) = n! / (r!(n-r)!)
Where:
- n is the total number of sales reps (10 in this case)
- r is the number of reps we want to choose for one territory (7 in this case)
- n! means the factorial of n, which is the product of all positive integers from 1 to n
Now, let's apply this formula to our problem:
C(10, 7) = 10! / (7!(10-7)!)
= 10! / (7!(3)!)
= (10 * 9 * 8 * 7!) / (7!(3 * 2 * 1))
= (10 * 9 * 8) / (3 * 2 * 1)
= 120
Therefore, there are 120 ways to assign 10 sales reps to two territories, with 7 reps in one territory and 3 in the other.