Question

in how many ways the letter HOME can b arranged so that no vowels come together​

Answer 1

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The permutation of 4 different letters taken 4 at a time (without any restriction) is

$\sf \mapsto {}^{4}P_{4} = \frac{4!}{(4 - 4)!} \\ \\ \sf \mapsto {}^{4}P_{4} = \frac{4 \times 3 \times 2 \times 1}{1} \\ \\ \sf \mapsto{}^{4}P_{4} = 24$

Let , the letters E and O be the single letter

So ,

The permutation of 3 different letters taken 3 at a time (when all vowels together) is

$\sf \mapsto {}^{3}P_{3} × 2! = \frac{3!}{(3- 3)!} \times 2! \\ \\ \sf \mapsto {}^{3}P_{3} × 2! = 3 \times 2 \times 1 \times 2 \times 1 \\ \\ \sf \mapsto {}^{3}P_{3} × 2! = 12$

Thus ,

The no. of arrangement = 24 - 12

The no. of arrangement = 12

Hence , the number of arrangement when all vowels do not occur together is 12

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