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 [2023]

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Solution

Q.
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 [2023]

1 5670

2 945

3 2835

4 1890

Ans.
(1)

In the word ''MATHEMATICS'' there are, $2 \to M's; 2 \to A's; 2 \to T's$

Total number of words formed by the letters of the word ''MATHEMATICS'' is $\frac{⌊ 11}{⌊ 2 \times ⌊ 2 \times ⌊ 2}$

Number of words in which 'C' and 'S' come together is $\frac{⌊ 10}{⌊ 2 \times ⌊ 2 \times ⌊ 2} \times 2$

Then, number of words in which 'C' and 'S' do not come together is $\frac{⌊ 11}{⌊ 2 \times ⌊ 2 \times ⌊ 2} - \frac{2 ⌊ 10}{⌊ 2 \times ⌊ 2 \times ⌊ 2} = \frac{9 \times ⌊ 10}{2 \times 2 \times 2}$

$\Rightarrow \frac{9 \times 10 \times 9 \times 8 \times 7 \times 6!}{2 \times 2 \times 2} = 6! \times k (∵ Given) \Rightarrow k = 5670$

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