Words of length 10 are formed using letters, A, B, C, D, E, F, G, H, I, J . Let x be number of such

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6.Permutation and Combination

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Words of length $10$ are formed using the letters, $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated ; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9 x}=$

A

$5$

B

$4$

C

$8$

D

$9$

(IIT-2017)

Solution

$\mathrm{x}=10!$

$\mathrm{y}={ }^{10} \mathrm{C}_1 \times{ }^9 \mathrm{C}_8 \times \frac{10!}{2!}$

$\Rightarrow \frac{\mathrm{y}}{9 \mathrm{x}}=\frac{{ }^{10} \mathrm{C}_1 \times{ }^9 \mathrm{C}_8}{9 \times 2!}=\frac{10 \times 9}{9 \times 2}=5$

$\Rightarrow 5$

Standard 11

Mathematics

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