SOLUTION: For positive integers n > r, show that: a) nCr = nC(n-r) b) nCr = (n-1)C(r-1) + (n-1)Cr

Question 1068818: For positive integers n > r, show that: a) nCr = nC(n-r) b) nCr = (n-1)C(r-1) + (n-1)Cr
Answer by swincher4391(1107) (Show Source): You can put this solution on YOUR website!
a) We know that the formula for nCr = = =
=
b) We know that the formula for nCr = = .
Usually proofs have some form of adding 0 or multiplying by 1 in a clever way. In this case we will represent n as n-r+r

Now distribute (n-1)!

Left hand fraction: cancel out n-r and combine r * (r-1)! to be r!
Right hand fraction: cancel out r and combine (n-r)*(n-r-1)! to be (n-r)!

Switch around n-r-1 to n-1-r, and we're basically done.

Rewrite n-r as (n-1)-(r-1) on the RHS fraction

RELATED QUESTIONS
Show that r(nCr) = n[(n-1)C(r-1)] (answered by Edwin McCravy)

Use the fact that nCr = n!/(n-r)!r! to show that: a.). nCr = nCn-r b.)... (answered by Edwin McCravy)

Prove that nC r + nCr-1 =... (answered by sachi)

prove that:... (answered by venugopalramana)

How to verify the identity of: (n+1)Cr= nCr +... (answered by ikleyn)

prove that:... (answered by venugopalramana)

Prove that... (answered by Alan3354,mathmate)

Let n and r be positive integers with n ≥ r. Prove that rCr + (r+1Cr)+...+( nCr) =... (answered by math_helper)

nCr/nCr-1 = (n-r+1)/r prove it (answered by Edwin McCravy)