<p>Find the number of permutations of all the letters of the word <strong>MATHEMATICS</strong> which starts with consonants only.</p>
Step-by-Step Solution
Key Concept: Identify that MATHEMATICS has 11 letters with specific repetitions (2 M's, 2 A's, 2 T's), and use complementary counting or direct counting by fixing a consonant in the first position, then arranging remaining 10 letters with adjusted repetitions.
<p><strong>Step 1:</strong> Identify letters in MATHEMATICS: M(2), A(2), T(2), H(1), E(1), I(1), C(1), S(1). Total 11 letters with consonants: M, T, H, C, S (5 consonants, with M and T repeated).</p><p><strong>Step 2:</strong> Use complementary approach: Total permutations - Permutations starting with vowels.</p><p><strong>Step 3:</strong> Total permutations of MATHEMATICS = $\dfrac{11!}{2! \times 2! \times 2!}$ (accounting for 2 M's, 2 A's, 2 T's)</p><p><strong>Step 4:</strong> Permutations starting with vowels (A, E, I): Fix a vowel in position 1. Remaining 10 letters contain the same repetitions except one A is used. Average case gives $4 \times \dfrac{10!}{2! \times 2! \times 2!}$ (A appears twice, E and I once each; fixing any reduces that letter's count).</p><p><strong>Step 5:</strong> Permutations starting with consonants = $\dfrac{11!}{2! \times 2! \times 2!} - \dfrac{4 \times 10!}{2! \times 2! \times 2!}$ = $\dfrac{11 \times 10! - 4 \times 10!}{2! \times 2! \times 2!}$ = $\dfrac{7 \times 10!}{2! \times 2! \times 2!}$</p><p>∴ Answer: $\dfrac{7 \times 10!}{2! \times 2! \times 2!}$</p>
Correct Answer: \(\dfrac{7 \times 10!}{2! \times 2! \times 2!}\)