Question

Six distinct integers are picked from the set {1,2,3,..10}, how many selections are there in which the second smallest integer in the group is 3?

A) 84
B) 56
C) 28
D) 21

Answer 1

To select 6 distinct integers from the set {1, 2, 3, ..., 10} such that the second smallest integer is 3, we can use combinations and permutations.

Step-by-step explanation:

To solve this problem, we can use combinations. We need to select 6 integers from the set {1, 2, 3, ..., 10} such that the second smallest integer is 3. Since 3 must be included, we need to select 5 integers from the remaining 9 integers (excluding 3). This can be done using combinations. The number of ways to select 5 integers from 9 is given by the combination formula: C(9, 5) = 9! / (5! * (9-5)!) = 126.

However, we need to consider that the integers picked must be distinct. So, we need to eliminate selections that include duplicate integers. For example, if we select 2, 3, 5, 5, 7, 9, this selection is not valid because it has duplicate integers. To eliminate these selections, we can use the concept of permutations.

The number of permutations of 6 distinct integers is given by P(10, 6) = 10! / (10-6)! = 151,200. Therefore, the number of valid selections is 126.