The given, formed word is of length 10.

It is given that x is the number of words where no letter is repeated.

Also, it is given that y is the number of words where exactly one letter is repeated twice and no other letter is repeated. Therefore,

x = 10!

and y = ¹⁰C₁ $$\times$$ ¹⁰C₂ $$\times$$ ⁹C₈ $$\times$$ 8!

Thus, $${y \over {9x}} = {{{}^{10}{C_1} \times {}^{10}{C_2} \times {}^9{C_8} \times 8!} \over {9 \times 10!}}$$

Using $${}^n{C_r} = {{n!} \over {r!(n - r)!}}$$, we get

$${y \over {9x}} = {{\left[ {{{10!} \over {1!(10 - 1)!}} \times {{10!} \over {2!(10 - 2)!}} \times {{9!} \over {8!(9 - 8)!}} \times 8!} \right]} \over {9 \times 10!}}$$

$$ = {{10!} \over {9 \times 2!\, \times 8!}} = {{10 \times 9 \times 8!} \over {9 \times 2 \times 8!}} = {{10} \over 2} = 5$$ [Using $$n! = n(n - 1)(n - 2)......1!$$]