Question

Find the number of permutations of all the letters of the word MATHEMATICS which starts with consonants only.

Answer 1

Hint: We have asked the number of permutations of an 11 letter word. We first need to identify the number of words repeated and for the number of times. Also, there is a condition so we need to find the number of consonants and find the permutations the word MATHEMATICS starts with only consonants.

Complete step-by-step answer:
We have been given the word MATHEMATICS and we need to find the number of permutations which starts with consonants only.
We need to understand that there are 11 letters in the word MATHEMATICS.
We need to identify the consonants first.
The consonants are M, T, H, C and S but from which M and T are repeated twice.
So we will have a separate formula for M and T.
Therefore, the number of ways the letters are arranged with M fixed is given by $\dfrac{10!}{2!\,\times 2!}$ .
There is a 10! In the numerator because there are 10 places remaining and there are 2! twice to avoid calculating the letter T and A because it appears twice.
Similarly, the number of ways the letters are arranged with T fixed is given by $\dfrac{10!}{2!\,\times 2!}$.
As H, C, S, are occurring once, therefore the number of ways the letters are arranged with H or C or s is given by $\dfrac{3\times 10!}{2!\,\times 2!\times 2!}$ .
We have multiplied 3 bcoz there are three letters for which this formula is applicable.
There is an additional 2! because now M is also calculated in the remaining alphabets.
Therefore, adding all the permutations we get,
The number of permutation of al the letters of the word MATHEMATICS is given by $\dfrac{10!}{2!\,\times 2!}+\dfrac{10!}{2!\,\times 2!}+\dfrac{3\times 10!}{2!\,\times 2!\times 2!}$.
Simplifying we get,
$\begin{align}
  & \dfrac{10!}{2!\,\times 2!}+\dfrac{10!}{2!\,\times 2!}+\dfrac{3\times 10!}{2!\,\times 2!\times 2!}=\dfrac{10!}{2!\,\times 2!}\left( 1+1+\dfrac{3}{2} \right) \\
 & =\dfrac{10!}{2!\,\times 2!}\left( 2+\dfrac{3}{2} \right) \\
 & =\dfrac{10!}{2!\,\times 2!}\left( \dfrac{7}{2} \right) \\
 & =\dfrac{7\times 10!}{8}
\end{align}$
Hence, the number of permutations of all the letters of the word MATHEMATICS which starts with consonants only are $\dfrac{7\times 10!}{8}$ .

Note: We need to take care that while calculating the permutation of the letter M and T there was 2! two times because there are either A or T repeating or A or M repeating. But for the letters H, C, S, all the three repeating letters M, T, A were present. Also, we need to take care that even though there are 11 letters we only took 10! because already one letter is fixed making the remaining letter to be 10.