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

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 ${=}^6{\mathrm{C}}_5\times 5!=6\times 120=720$.

(B) 2 alike, 3 different

Number of ways ${=}^3{\mathrm{C}}_1{\times }^5{\mathrm{C}}_3\times \frac{5!}{2!}=3\times 10\times 60=1800$

(C) 3 alike, 2 different

Number of ways ${=}^2{\mathrm{C}}_1{\times }^5{\mathrm{C}}_2\times \frac{5!}{3!}$

= 2 × 10 × 20 = 400

(D) 2 alike, 2 alike, 1 different

${=}^3{\mathrm{C}}_2{\times }^4{\mathrm{C}}_1\times \frac{5!}{2!2!}=3\times 4\times 30=360$

(E) 3 alike, 2 alike ${=}^2{\mathrm{C}}_1{\times }^2{\mathrm{C}}_1\times \frac{5!}{3!2!}$

= 2 × 2 × 10 = 40

Hence, total number of ways

= 720 + 1800 + 400 + 360 + 40 = 3320.