First slide

Combinations

Question

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

Moderate

A

4200
B

3320
C

3840
D

None of these

Solution

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 × 10 × 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 × 2 × 10 = 40
Hence, total number of ways
= 720 + 1800 + 400 + 360 + 40 = 3320.

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