Bagaimana membuktikan nCr + nCr – 1 = n + 1Cr?

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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