if m = nc2 , prove that mc2 = 3n+1c4 - Maths - Binomial Theorem

Question

if m = nc2 , prove that mc​2 =
3n+1c4

Ajanta Trivedi · Accepted Answer

We have,      m=C2n⇒m = n!n-2! 
 2!⇒m = nn-1n-2!n-2! 
 2⇒m=n(n-1)2
Now, LHS = C2m=m(m-1)2=n(n-1)2 n(n-1)2-12=n(n-1)
n(n-1)-28=n(n-1)[n2-n-2]8=n(n-1)[n2-2n+n-2]8=n(n-1)[n(n-2)+1(n-2)]8=n(n-1)(n-2)(n+1)8
=3×(n+1) n(n-1) (n-2)3 2 4=3×(n+1)
n(n-1)(n-2)4!Now, RHS = 3C4n+1=3 × n+1!n+1-4! 
 4!=3 × n+1nn-1n-2n-3!n-3! 
 4!=3 × nn+1n-1n-24!So, LHS = RHS

hope this helps you

if m = nc2 , prove that mc​2 = 3n+1c4

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