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Question: There are 10 pairs of shoes on a cup board. The no. of ways can 4 shoes be selected so that there is...

There are 10 pairs of shoes on a cup board. The no. of ways can 4 shoes be selected so that there is no pair is

A

10C4.2410C_{4}.2^{4}

B

10C410C_{4}

C

10C6.2610C_{6}.2^{6}

D

10C510C_{5}

Answer

10C4.2410C_{4}.2^{4}

Explanation

Solution

Method I: No. of ways selecting 4 pairs = 10C4.

From each pair we can select left or right shoe.

For each pair there are 2 choices.

For 4 pairs we have 24 choices.

∴ Total no. of ways = 10C4.24.

Method II: = 10C2.8C2+210C3.7C1+210C410C_{2}.^{8}C_{2} + 2^{10}C_{3}.^{7}C_{1} + 2^{10}C_{4}.

Method III: = 20.18.16.14∠4\frac{20.18.16.14}{\angle 4}