Question

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

• (picture #1)

The (n+ 1)th row of this table gives the coefficients for 

(a+b)^n = ∑^{^n}_{r=0  n}C_k  a^rb^n-r

The next row is found by adding the two numbers above the new entry, i.e.

• (picture #2)

Prove this equation using the mathematical definition of a combination.!!!!!!

Answer 1

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!]