Question

Find the number of four letter words that can be formed from the letters of the word ${}^{\prime }{\text{PROPORTION}}^{\prime }$.

• A. $740$
• B. $758$
• C. $700$
• D. $624$

Answer 1

Answer: (B)

$758$

$PROPORTION$

$P⟶2$

$R⟶2$

$O⟶3$

$T⟶1$

$I⟶1$

$N⟶1$

Case $:1⟶6$ diffrent letters

Number of ways of arranging ${=}^4{C}_2\times 4!=360$ Ways

Case $:2⟶$ Words with exactly a letter repeated twice

${=}^3{C}_1=3$ Ways

All other two distinct letters ${=}^5{C}_2=10$ Ways

Each combination $=\frac{4!}{2!}=12$ Ways

Total possible words $=3\times 10\times 12=360$ Ways

Case $:3$ Two distinct letters repeated twice

${}^3{C}_1=3$

Each combination $=\frac{4!}{2!\times 2!}=6$

Total words $=3\times 6=18$ Ways

Words with exactly a letter repeated thrice

Selecting $1$ letter out of $5$ remaining options

${=}^5{C}_1=5$ Ways

Each combintion

$=\frac{4!}{3!}=4$

Total number of such words $=1\times 5\times 4=20$ Words

Total Number of overall arrangements

$=360+360+18+20=758$ Ways