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.