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

Answer 1

Answer: (D)

In the word MONDAY, all letters are different.
I. Out of 6 different letters, 4 letters can be selected in ${ }^6 P_4$ ways.
$\therefore $ 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 $={ }^6 P_6$
$\therefore$ 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 $A$.
$\therefore$ First letter can be chosen by 2 ways.
Number of ways taking 5 different letters from remaining 5 letters $={ }^5 P_5$
$\therefore$ 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 $=2 \times 120=240$