Question

The number of words that can be formed by using the letters of the word MATHEMATICS that start as well as end with T are

1. \(\dfrac{9!}{2!\times 2! \times 2!}\)
2. 9!
3. \(\dfrac{9!}{2!\times 2!}\)
4. None of these

Answer 1

Correct Answer - Option 3 : \(\dfrac{9!}{2!\times 2!}\)

Concept:

Suppose a set of n objects has n₁ of one kind of object, n₂ of a second kind, n₃ of a third kind, and so on with n = n₁ + n₂ + n₃  +...+ n_k then the number of distinguishable permutations of the n objects is  = \(\rm \frac{n!}{n_1!n_2! ...n_k!}\)

 

Calculation:

The given word is MATHEMATICS

The total number of letters is 11.

Given that we have to form those words which start and end with T.

T _ _ _ _ _ _ _ _ _ _ T

 

Since there are two identical letters and have to be placed in two positions so we can place it only 1 way.

Now we have the remaining 9 positions and the remaining 9 letters are M, A, H, E, M, A, I, C, S

Now in the given word, there are 2T’s, 2M’s and 2A’s, from where 2T’s are already used, so since there are 2M’s and 2A’s

Therefore, the total number of words = \(\dfrac{9!}{2!\times 2!}\)