How to show?

Question

Show that ⁿC_r = ⁿC_n-r .Without the use of tables or a calculator, evaluate ²⁰C₃.

Answer 1

nCr = n! / [r! (n-r)!] by definition.

 

now try substituting (n-r) for r in the above, and simplify, and you'll have the proof.

 

20C3 = 20! / [17! 3!]  and the 17! cancels the tail end of 20!, leaving only 20(19)(18) / 3! and this becomes 20(19)(3) and you can finish the evaluation.

 

Note that  nCr = nCn-r   is consistent with the fact that Pascal' triangle, which gives binomial coefficient values, has every line reading the same whether left to right ot R-to-L reading.