JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Permutations and Combinations

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If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is

(6!) k

, then k is equal to

(A)1890
(B)945
(C)2835
(D)5670✓ Correct

Answer: (D)

Solution

The word MATHEMATICS has 11 letters: M(2), A(2), T(2), H(1), E(1), I(1), C(1), S(1).

First arrange the 9 letters (excluding C and S):

\frac{9 !}{2 ! \cdot 2 ! \cdot 2 !}

These 9 letters create 10 gaps (including ends). Place C and S in any 2 of these 10 gaps:

^{10} C_{2} \cdot 2! = 45 \cdot 2 = 90

Total arrangements:

\frac{9 !}{2 ! \cdot 2 ! \cdot 2 !} \cdot 90 = \frac{362880}{8} \cdot 90 = 45360 \cdot 90 = 4082400

Given this equals

(6!) k = 720 k

k = \frac{4082400}{720} = 5670

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Single correct option

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