=10 ⇒5r=120 ⇒r=24 ∴ Coefficient of x10=60C24⋅5(
3×24−60
4
) =60C24⋅53 =
60!
24!36!
⋅53 It is given that,
60!
24!36!
⋅53=5k⋅l...... (1) Also given that, I is coprime to 5 means I can't be multiple of 5 . So we have to find all the factors of 5 in 60!,24 ! and 36 ! [Note : Formula for exponent or degree of prime number in n!. Exponent of p in n!=⌈
n
p
⌋+⌈
n
p2
⌋+[
n
p3
]+..... until 0 comes here p is a prime number. ] ∴ Exponent of 5 in 60 ! =⌈
60
5
⌋+⌈
60
52
⌋+⌈
60
53
⌋+...... =12+2+0+.... =14
Exponent of 5 in 24 ! =⌈
24
5
⌋+⌈
24
52
⌋+⌈
24
53
⌋+...... =4+0+0...... =4 Exponent of 5 in 36 ! =⌈
36
5
⌋+⌈
36
52
⌋+⌈
36
53
⌋+...... =7+1+0...... =8 ∴ From equation (1), exponent of 5 overall
