Question

How many words, with or without meaning can be made from the letters of the word $MONDAY$ assuming that no letter is repeated, if
$\left(i\right)4$ letters are used at a time.
$\left(ii\right)$ All letters used at a time.
$\left(iii\right)$ All letters are used but first letter is a vowel.

Answer 1

$\left(i\right)$ Finding total number of alphabets with $4$ letters used at a time and without repetition.
Total number of alphabets in $MONDAY=n=6$
$4$ letters are used at a time, so $r=4$
$\therefore$ number of different words $={}^n{P}_r={}^6{P}_4$
$=\frac{6!}{(6-4)!}=\frac{6\times 5\times 4\times 3\times 2\times 1}{2\times 1}=360$

$\left(ii\right)$ Finding total number of words when all letters used at a time.
Total number of alphabets in $MONDAY=n=6$
all letters are used at a time, $\Rightarrow r=6$
$\therefore$ number of different words $={}^n{P}_r={}^6{P}_6$
$=\frac{6!}{(6-6)!}=\frac{6\times 5\times 4\times 3\times 2\times 1}{0!}=720$

$\left(iii\right)$ Finding total Number of words when all letters are used but first letter is a vowel.
Vowels in $MONDAY$ are $O$ and $A$
Number of ways to fill first place with an vowel
${=}^2{P}_1=2$

After first place is filled, remaining places should be filled with remaining $5$ letters.
Number of ways to fill remaining five places with $5$ letters
$={}^5{P}_5=5!=120$
$\therefore$ Number of words when all letters are used but first letter is a vowel $=120\times 2=240$.