Question

Prove the below statement for n>=2 and 1 <= j <=n

2^n >= (n(n-1)...(n-j+1))/j!

Please explain with a detailed proof, thanks

Answer 1

We have to prove O z n(n-1)., (najt) 15jen, nz2 We shall prove the result by principle of mothematical induction For n=2 Here ja 1 or a = 4 Rais a mint) (naj4) = m (na)., (n=(1-) For j=1 RHS = 22,

For j=1 R. Hs= 2 = 2 . For j=2 RHS= 2(1) - 1 ii. 4>2,1 result is true for For n=3 LHS = 23 = 6 Here j = 1, 2, 3 For j=1 RHS = 3 = 3 n=2 For j=2 Rits= 3(2) 3 for j =3 Rhs - 3(2)(1) 21 31 23> nint... (n-j+1) f. 1 = 3 = 3

Result is true for n=3 also Assume that the result is true for n=k e K 2 K (Ku) (K-2) --- CK-1+1). 14,<k we shall prove the result for n=kt! Also

LHS K+ For RHS j= 1, 2, ...K+ Obviously 25t..> K+! =(:') For je! RHis = (+1) 682) = kt] For ju? RHS = (K +) K . . For jaz Rusa (xal) K (ku) kt! K(K+1) =(K*'). "> (K-) ke(ka) = (K3V) - 1. KAI 27 3! , Continuing For jakt, in this way we have RHS = (K+1) Ck) *K1) ... 3.2.1 - 1 2x+1)! Thus LHS 7 RHS Therefore result is Noter we know for n=k+1 true for all i na also n a K +1 DK+! so we can eally write 2* any Y=1,2,.. (K+ 1 for t hanks