 # How many words can be formed from the letters of the word ‘SUNDAY’?

Question:

How many words can be formed from the letters of the word ‘SUNDAY’? How many of these begin with D?

Solution:

There are 6 letters in the word SUNDAY.

Different words formed using 6 letters of the word SUNDAY is $P(6,6)$

Formula:

Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is

$P(n, r)=n ! /(n-r) !$

Therefore, a permutation of 6 different objects in 6 places is

$P(6,6)=\frac{6 !}{(6-6) !}=\frac{6 !}{0 !}=\frac{720}{1}=720$

720 words can be formed using letters of the word SUNDAY.

When a word begins with $D$.

Its position is fixed, i.e. the first position.

Now rest 5 letters are to be arranged in 5 places.

Formula:

Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is

$P(n, r)=n ! /(n-r) !$

Therefore, a permutation of 5 different objects in 5 places is

$P(5,5)=\frac{5 !}{(5-5) !}=\frac{5 !}{0 !}=\frac{\frac{120}{1}}{1}=120$

Therefore, the total number of words starting with D are 120.