PROBLEM 12E

Prove

Hint: Consider (1 − 1)n and (1 + 1)n, or use Pascal’s equation and proof by induction.

Solution 12E

Step1 of 3:

We need to prove,

nCr =0 and =

Consider (1 − 1)n and (1 + 1)n, or use Pascal’s equation and proof by induction.

Step2 of 3:

Consider,

nCr = 0

Proof: we know that the pascal’s equation is given by

nCr

Substitute a = -1 and b = 1 in above equation we get

nCr

= nCr

0 = nCr

0 = nCr{}

0 = nCr{}

0 = nCr

Therefore, nCr = 0

Hence the proof.

Step2 of 3:

Again consider,

=

Proof: we know that the pascal’s equation is given by

nCr

Substitute a = 1 and b = 1 in above equation we get

nCr

= nCr

= nCr

= nCr{}

= nCr{}

= nCr

Therefore, nCr =

Hence the proof.