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Of all the 8 letter words that can be formed by using all letters of the word EQUATION. How many begin with the letter T?
(a) $8!$
(b) $\dfrac{7!}{2!}$
(c) $7!$
(d) $6!$

Last updated date: 24th Jul 2024
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Hint: Form 8 boxes and fix the letter T in the first box. Now, fill the remaining 7 boxes with 7 letters that are remaining by using the formula that n number of different things can be arranged in $n!$ ways. Substitute n = 7 to get the answer.

Complete step by step answer:
Here we are provided with the word EQUATION and we are asked to find the number of 8 letters words that can be formed using the letters present in the word such that the first letter is always T.
Now, we can see that there are 8 different letters in the given word so let us form 8 boxes in which the letters can be filled. The first letter should be T as per the condition given in the question, so let us fill the first box with T. Therefore we have,

Now, we need to fill the remaining 7 boxes with 7 remaining letters of the word. We know that n things can be arranged in $n!$ ways if there are n positions. So we have,
$\Rightarrow $ Number of ways in which the remaining 7 letters of the given word can be arranged in 7 boxes = $7!$
So, the correct answer is “Option c”.

Note: Note that here none of the letters were repeating or appearing more than once. In case the letters appear more than once, we divide the total number of arrangements (considering all the letters different) with the product of factorial of number of times each letter is repeated. We use the formulas of permutations and combinations for the processes of arrangement and selection respectively.