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

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

Permutations and Combinations

Solution:

There are $11$ letters in the word INDEPENDENT. $3 N , 3 E , 2 D , I , P , T$ ($6$ 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 $