JEE Advanced · 2017 · 21 May 2017 · Paper 1 · Permutations and Combinations

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} =

Answer: 5

Solution

For words with no repetition:

x = 10!

. For words with exactly one letter repeated twice: choose which letter repeats (

^{10} C_{1}

ways), choose 8 other letters from the remaining 9 (

^{9} C_{8}

ways), and arrange these 10 letters where 2 are identical (

\frac{10 !}{2 !}

ways). Thus

y =^{10} C_{1} \times^{9} C_{8} \times \frac{10 !}{2 !} = 10 \times 9 \times \frac{10 !}{2}

. Therefore

\frac{y}{9 x} = \frac{10 \times 9 \times \frac{10 !}{2}}{9 \times 10 !} = \frac{10}{2} = 5

JEE Advanced · 2017 · 21 May 2017 · Paper 1 · Integer answer