Bagaimana membuktikan nCr + nCr – 1 = n + 1Cr?
nCr = n! / (nr)! xr! dan r! = rx (r-1)!
nCr
= n! / (nr)! xr!
= n! / (nr)! xr(r-1)!
nCr-1
= n! / (n-(r-1))! x (r-1)!
= n! / (n-r+1)! x (r-1)!
= n! / (n-r+1)(nr)! x (r-1)!
(n+1)Cr
= (n+1)! / ((n+1) – r)! xr!
= (n+1)n! / (n-r+1)! xr(r-1)!
sekarang
LHS
= nCr + (n+1)Cr
= n! / (nr)! xr(r-1)! + n! / (n-r+1)(nr)! x (r-1)!
= n! ( (n-r+1) + r / (n-r+1)(nr)!xr(r-1)! )
= n! ( n-r+1-r / (n+1 – r)(nr)! xr! )
= n! ( n+1 / ((n+1) – r)! xr! )
= (n+1)n! / ((n+1) -r)! xr!
= (n+1)! / ( (n+1) -r )! xr!
= (n+1) C r
= RHS
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