Question

The total number of ways of selecting five letters from the letters of the word INDEPENDENT, is

• A. 4200
• B. 3320
• C. 3840
• D. None of these

Answer 1

Answer: (B)

There are $11$ letters in the word INDEPENDENT. $3N,3E,2D,I,P,T$ ($6$ types)
Different possibilities of choosing 5 letters are
(A) All different
Number of ways $={}^6{C}_5\times 5!=6\times 120=720$.
(B) $2$ alike, $3$ different
Number of ways $={}^3{C}_1\times {}^5{C}_3\times \frac{5!}{2!}=3\times 10\times 60=1800$
(C) $3$ alike, $2$ different
Number of ways $={}^2{C}_1\times {}^5{C}_2\times \frac{5!}{3!}$ $=2\times 10\times 20=400$
(D) $2$ alike, $2$ alike, $1$ different
$={}^3{C}_2\times {}^4{C}_1\times \frac{5!}{2!2!}=3\times 4\times 30=360$
(E) $3$ alike, $2$ alike $={}^2{C}_1\times {}^2{C}_1\times \frac{5!}{3!2!}$ $=2\times 2\times 10=40$
Hence, total number of ways
$=720+1800+400+360+40=3320$