Prove that     n C r + n C r-1 =  n+1 C r

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nathanshields | High School Teacher | (Level 1) Associate Educator

Posted on

Hi - looking at these equations can be a little confusing, so just take them one at a time, writing down each step to make sure you understand.

`nCr=(n!)/(r!(n-r)!)`

`nCr-1=(n!)/((r-1)!(n-r+1)!)`

Adding them together requires a common denominator, which can be r!(n-r+1)!

`(n!)/(r!(n-r)!)*(n-r+1)/(n-r+1)+(n!)/((r-1)!(n-r+1)!)*r/r`

`=(n!(n-r+1)+n!r)/(r!(n-r+1)!)`

`=(n!(n-r+1+r))/(r!(n+1-r)!)`

=`(n!(n+1))/(r!(n+1-r)!)`

=`((n+1)!)/(r!(n+1-r)!)`

Cha-ching!  This is what we wanted! ```(n+1)C(r)`

Thanks,
SHIELDS

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