Question

Find number of ways in which an arrangement of 4- letters can be made from the letter of the word PROPORTION.

• A. $758$
• B. $753$
• C. $751$
• D. $759$

Answer 1

Answer: (A)

$758$

$P\to 2$

$R\to 2$

$O\to 3$

$T\to 1$

$I\to 1$

$N\to 1$

Case 1 : words with 4 distinct letters.

we have 6 letters & 4 places ${}^6{C}_4\times 4!=360$ ways

Case 2 : Words with exactly a letter repeated twice

we have P, R, O repeating itself we have to chore any 1

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

other words can be select

${}^5{C}_2=10ways$

Now each combination can be arranged $\frac{4!}{2!}=12$ ways

So, total no. of such words = $3\times 10\times 12=360$

Case 3 : Words with exactly 2 distinct letter repeat

$P,R,O\to {}^3{C}_2=3$ ways

Each can be select = $\frac{4!}{2!2!}=6$

5 total ways = $3+6=18$

Case 4 : $3\to$ at a time

$0\to 1$

for other place = ${}^5{C}_1=5$

Total = $1\times 5\times \frac{4!}{3!}=20$

Total = $360+360+18+20=758$ ways