Question

The numbers of arrangements that can be made by taking $4$ letters at a time, out of the letters of the word ${}^{\prime }PASSPOR{T}^{\prime }$ is

• A. $606$
• B. $4464$
• C. $4356$
• D. $4644$

Answer 1

The correct option is A $606$

$PASSPORT$

Total letters $=8$

$2P,2S$, and single $\{A,O,R,T\}$

We have to select $4$ letters from the word $PASSPORT$

Case $1:$ Two alike letters and other two alike

from $\left\{PPSS\right\}$ total $4$ letter arrangements $=\frac{4!}{2!\times 2!}=6$ ways ………………$\left(I\right)$

Case $2:$ Two alike letter and other two different letters

one alike letter from $\{P,S\}$ and two different from $\{P,S,A,O,R,T\}$

Total combinations ${=}^2{C}_1{\times }^5{C}_2=20$ ways

total arrangements $=20\times \frac{4!}{2!}=240$ ways …………………$\left(II\right)$

Case $3:4$ different letters from set $\{P,S,A,O,R,T\}$

Total combinations ${=}^6{C}_4=15$ ways

Total arrangements $=15\times 4!=360$ ways …………………………$\left(III\right)$

So Final Arrangements $=\left(I\right)+\left(II\right)+\left(III\right)=6+240+360=606$