Prove that Hint Pascal Triangle 2Using the previous two prob

Prove that

Hint: Pascal Triangle!

2-Using the previous two problems, find all positive integers a, b, c such that

Let a b c be non-negative integers. Prove that Find all non-negative integers n r such that Using the previous two problems, find all positive integers a, b, c such that

Solution

1.

We first express the right side in terms of factorials:

C(a, c) C(a-c,b-c)

= {a! / [c!(a - c)!] } { (a - c)! / [(b - c)! (a - b)!]

Cancelling (a - c)!,

= {a! / [c!] } { 1 / [(b - c)! (a - b)!]

= a! / [c! (b - c)! (a - b)!] [1]

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Now, for the left side, in terms of factorials:

C(a,b) C(b,c)

= {a! / [b! (a-b)!]} { b! / [c! (b - c)!] }

Cancelling b!,

= {a! / [(a-b)!]} { 1/ [c! (b - c)!] }

Rearranging terms,

= = a! / [c! (b - c)! (a - b)!] [2]

Which is the same as expression [1] above.

DONE!

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