The total number of $7$ -letter words that can be formed using the letters of the word $M A T H E M A T I C S$ such that the repeating letters should be placed in odd places, whereas the non-repeating letters should be placed in even places, is

Open in App

Solution

There are $4$ odd places and $3$ even places in a $7$ -letter word.
Repeating letters $- M M, A A, T T$
Non-repeating letters $- H, E, I, C, S$
Even places can be filled in $^{5} P_{3} = 60$ ways.

For odd places, there are two cases :
$(i)$ Two letters out of four are identical and two are different.
Number of ways $=^{3} C_{1} \times \frac{4!}{2!} = 36$

$(i i)$ Two letters are identical to each other and other two are also identical to each other.
Number of ways $=^{3} C_{2} \times \frac{4!}{2! 2!} = 18$
So, total number of ways to fill even places $= 36 + 18 = 54$

Hence, total number of possible words is $54 \times 60 = 3240$

Suggest Corrections

Similar questions

Q. The total number of

7

-letter words that can be formed using the letters of the word

M A T H E M A T I C S

such that the repeating letters should be placed in odd places, whereas the non-repeating letters should be placed in even places, is

A five letter word is to be formed such that the letters appearing in the odd positions are taken from the unrepeated Letters of the word MATHEMATICS whereas the letters which occupy even places are taken from amongst the repeated letters. Then number of such words is

Q. A five letter word is to be formed by using the letters of word MATHEMATICS such that
(i) odd places of the word are to be filled with unrepresented letters and
(ii) even places are to be filled with repeated letters.
Then the number of words thus formed is:

Q. A five letter word is to be formed such that the letters appearing in the old position are taken from the unrepeated letters of the word MATHEMATICS whereas the letters which occupy even places are taken from amongst the repeated letters.

Q. The number of words that can be formed using any number of letters of the word "KANPUR" without repeating any letter is

Join BYJU'S Learning Program

The total number of 7-letter words that can be formed using the letters of the word MATHEMATICS such that the repeating letters should be placed in odd places, whereas the non-repeating letters should be placed in even places, is

The total number of $7$ -letter words that can be formed using the letters of the word $M A T H E M A T I C S$ such that the repeating letters should be placed in odd places, whereas the non-repeating letters should be placed in even places, is