Pascals triangle gives a method for calculating the binomial

Pascal’s triangle gives a method for calculating the binomial coefficients; it begins as follows:

The nth row of this triangle gives the coefficients for (a + b)n?1. To find an entry in the table other than a 1 on the boundary, add the two nearest numbers in the row directly above. The equation

called Pascal’s equation, explains why Pascal’s triangle works. Prove that this equation is correct.

Solution

Note that

nCr = n! / [(n - r)! r!]

(n - 1)Cr = (n-1)! / [(n - r - 1)! r!]

(n - 1)C(r - 1) = (n - 1)! / [(n - r)! (r - 1)!]

Thus,

(n - 1)Cr + (n - 1)C(r - 1) = (n-1)! / [(n - r - 1)! r!] + (n - 1)! / [(n - r)! (r - 1)!]

Multiplying both terms on the right by n/n, the (n - 1)! become n!,

(n - 1)Cr + (n - 1)C(r - 1) = n! / [(n - r - 1)! r! n] + n! / [(n - r)! (r - 1)! n]

Putting them together as one fraction, as their LCD is (n - r)! r! n,

(n - 1)Cr + (n - 1)C(r - 1) = n! [(n - r) + (r)] / [(n - r)! r! n]

(n - 1)Cr + (n - 1)C(r - 1) = n! [n] / [(n - r)! r! n]

(n - 1)Cr + (n - 1)C(r - 1) = n! / [(n - r)! r!] = nCr [DONE!]