Six distinct integers are picked at random from {1,2,3,…,10}. What is the probability that, among those selected, the second smallest is 3?

1986 AMC 12 Problem 22

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22.

Six distinct integers are picked at random from

\{1,2,3,\ldots,10\}.

What is the probability that, among those selected, the second smallest is

3?

\frac1{60}

\frac16

\frac13

\frac12

none of these

Count all six-element subsets of the ten integers

For second-smallest

3,

choose one element below

3

and four above

3

There are

\binom{10}{6}=210

possible sets. If the second-smallest element is

3,

then

3

is selected, one element is chosen from

\{1,2\},

and four are chosen from

\{4,5,\ldots,10\}.

This gives

2\binom74=70

favorable sets. The probability is

\frac{70}{210}=\frac{1}{3}.

Thus the correct answer is C.