The number of words with or without meaning that can be made form the letters of the word MONDAY, assuming that no letter is repeated, if Statement I 4 letters are used at a time equals 360. Statement II All letters are used at a time equal 720. Statement III All letters are used but first letters is a vowel equal 120. Identify the correct combination of true (T) and false (F) of the given statements.

Question

The number of words with or without meaning that can be made form the letters of the word MONDAY, assuming that no letter is repeated, if Statement I 4 letters are used at a time equals 360. Statement II All letters are used at a time equal 720. Statement III All letters are used but first letters is a vowel equal 120. Identify the correct combination of true (T) and false (F) of the given statements. (A) TTT (B) FFF (C) FFT (D) TTF

Tardigrade Admin · Accepted Answer

In the word MONDAY, all letters are different.
I. Out of 6 different letters, 4 letters can be selected in

^{6} P_{4}

ways.

∴

Required number of words

=^{6} P_{4}

= \frac{6 !}{( 6 - 4 )!} = \frac{6 !}{2 !}

= \frac{6 \times 5 \times 4 \times 3 \times 2 !}{2 !}

= 360

II. The word 'MONDAY' has 6 different letters.
Number of ways of arranging 6 letters at a time

=^{6} P_{6}

∴

Required number of words

=^{6} P_{6}

= \frac{6 !}{( 6 - 6 )!}

= \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{0 !}

= 720

III. First, we will fix the vowels.
In the word 'MONDAY', there are two vowels

O

and

A

.

∴

First letter can be chosen by 2 ways.
Number of ways taking 5 different letters from remaining 5 letters

=^{5} P_{5}

∴

Required number of words

=^{5} P_{5} = \frac{5 !}{( 5 - 5 )!} = \frac{5 !}{0 !}

= 5 \times 4 \times 3 \times 2 \times 1 = 120

Hence, by multiplication rule, the total number of ways

= 2 \times 120 = 240