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How many different selections of 4 books can be made from 10 different books, if
(i) there is no restriction

(ii) two particular books are always selected

(iii) two particular books are never selected

1 Answer

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Best answer

Given as

The total number of books = 10

The total books to be selected = 4

(i) there is no restriction

The number of ways = choosing 4 books out of 10 books

10C4

On using the formula,

nCr = n!/r!(n – r)!

10C4 = 10! / 4! (10 – 4)!

= 10! / (4! 6!)

= [10 × 9 × 8 × 7 × 6!] / (4! 6!)

= [10 × 9 × 8 × 7] / (4 × 3 × 2 × 1)

= 10 × 3 × 7

= 210 ways

(ii) two particular books are always selected

The number of ways = select 2 books out of the remaining 8 books as 2 books are already selected.

8C2

On using the formula,

nCr = n!/r!(n – r)!

8C2 = 8! / 2! (8 – 2)!

= 8! / (2! 6!)

= [8 × 7 × 6!] / (2! 6!)

= [8 × 7] / (2 × 1)

= 4 × 7

= 28 ways

(iii) two particular books are never selected

The number of ways = select 4 books out of remaining 8 books as 2 books are already removed.

8C4

On using the formula,

nCr = n!/r!(n – r)!

8C4 = 8! / 4! (8 – 4)!

= 8! / (4! 4!)

= [8 × 7 × 6 × 5 × 4!] / (4! 4!)

= [8 × 7 × 6 × 5] / (4 × 3 × 2 × 1)

= 7 × 2 × 5

= 70 ways

Hence, the required no. of ways are 210, 28, 70.

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