20 pairs of shoes are there in a closet. Four shoes are selected at random. The probability that they are pairs is

\frac{^{20} C_{4}}{^{40} C_{4}}

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\frac{^{20} C_{2}}{^{40} C_{4}}

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\frac{^{20} C_{2}}{^{20} C_{2}}

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\frac{^{20} C_{4}}{^{20} C_{4}}

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Solution

The correct option is B $\frac{^{20} C_{2}}{^{40} C_{4}}$
For me to choose a pair, I just have to choose a shoe and I lose out on one choice as it has to be the corresponding pair.
So, it is as if I just have the choice to choose a pair.
Out of 20 given pairs, the number of ways in which I can choose a pair is $^{20} C_{2}$
The Universal set, or the total number of choices I have to take out 4 shoes is $^{40} C_{4}$
The probability is $\frac{^{20} C_{2}}{^{40} C_{4}}$

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