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)

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 $={}^6{C}_2\times 2!={}^6{P}_2$