Bunuel
Ashley and Vinnie work on a sales staff with 8 other salespeople. If five members of the staff will be chosen at random to attend a sales workshop, what is the probability that both Ashley and Vinnie will be chosen?
A. 1/10
B. 3/28
C. 2/9
D. 1/4
E. 1/2
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:This problem includes a mix of combinatorics and probability. The probability that both Ashley and Vinnie will be chosen is: the number of teams on which they're both chosen divided by the number of total possible teams. In these situations it's usually easier to start with the total. With 10 total people and 5 chosen, the number of combinations (order doesn't matter!) can be calculated as:
10!/(5!5!), which factors out to (10)(9)(8)(7)(6)/[(5)(4)(3)(2)]. That ultimately reduces to 252, but remember that you're doing all of this to find a denominator, the number of total teams. So you may want to leave the math undone; as you factor the numerator 6 with the denominator 2 and 3, and then divide the 10 by 5 and the 8 by 4, you're really left with: 2 * 9 * 2 * 7, which you'll find makes an easier denominator. Then for the number of teams that both Vinnie and Ashley are on. The logic here: if those two are on the team, then there are 8 people left to be chosen for 3 remaining spots. So the calculation is: 8!/(3!5!), which reduces to (8)(7)(6)/[(3)(2)], which is 56 (or you can just leave the 8 * 7 undone). That leaves you with a "teams they're on, divided by teams total" calculation of: (8)(7)/[(2)(2)(9)(7)], in which the 7s cancel and the 8 divided by (2 * 2) leaves a 2 in the numerator. That simplifies to 2/9.