Question

I. The no.of ways in which 4 letters can be posted in 5 letter boxes in $4^5$ ways.
II. If words are formed by taking only 4 at a time out of the letters of the word PHYSICS, then the number of words in which Y occurs, is 288.
Which of the above statements is correct ?

• A. only I
• B. only II
• C. both I and II
• D. Neither I nor II

Answer 1

The correct option is B only II

Solution for part 1 is $\left(\begin{array}{c}5\\ 1\end{array}\right)\left(\begin{array}{c}5\\ 1\end{array}\right)\left(\begin{array}{c}5\\ 1\end{array}\right)\left(\begin{array}{c}5\\ 1\end{array}\right)=5^4$
and
Solution for part 2 is one letter Y is always taken.
Hence total number of ways $=$ (All different along with Y)+ (Two same along with Y)
=$\left(\begin{array}{c}5\\ 3\end{array}\right)\times 1\times 4!+\left(\begin{array}{c}4\\ 1\end{array}\right)\times \left(\begin{array}{c}2\\ 2\end{array}\right)\times 1\times \frac{4!}{2!}=240+48=288$
Hence statement 1 is wrong and statement 2 is correct.