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$, y/(9x) =$ ______

$\text{Letters: } A,B,C,D,E,F,G,H,I,J\ \ (10 \text{ letters}),\ \text{length } 10$ $x = \#\{\text{no letter repeated}\}$ $y = \#\{\text{exactly one letter twice, others distinct}\}$ $\text{To find: } \dfrac{y}{9x}$ $x = {}^{10}P_{10} = 10!$ $\text{choose the repeated letter: } 10 \text{ ways}$ $\text{choose the other 8 distinct letters from remaining 9: } {}^{9}C_{8} = 9$ $\text{arrange: } \dfrac{10!}{2!} = \dfrac{10!}{2}$ $y = 10 \times 9 \times \dfrac{10!}{2} = 45 \times 10!$ $\dfrac{y}{9x} = \dfrac{45 \times 10!}{9 \times 10!} = \dfrac{45}{9} = 5$