Question

The total number of $3$ letter words that can be formed from the letter of the word $SAHARANPUR$

• A. when all the letters are different is $210$
• B. when $2$ letters are alike and $1$ is different is $30$
• C. when all the letters are alike is $1$
• D. when the letter starts with $A$ and all the letters are different is ${}^6{C}_2$

Answer 1

Answer: (A), (C)

The correct options are
A when all the letters are different is $210$
C when all the letters are alike is $1$

Given word consists $1S,3A,1H,2R,1N,1P,1U$
Now,when all the three letters are different i.e.,$XYZ$ type corresponding words will be
${}^7{P}_3=210$

When two letters are alike and other is different i.e.,$XXY$ corresponding number of ways is
${}^2{C}_1\times {}^6{C}_1\times \left(\frac{3!}{2!}\right)=36$.

When all letters are alike i.e.,$XXX$ type, corresponding number of ways is $1$.

Thus, total number of words that can be formed is
$210+36+1=247$.

When the letter starts with $A$ and letters are unique
The remaining letters can be selected in${}^6{C}_2$ ways and arranged in $2!$ ways
Total ways ${}^6{C}_2\times 2!={}^6{P}_2$