Prove that `underset (r=1) overset (5) sum "^5C_r=31`.

Prove that underset (r=0) overset (n) sum 3^r ""^nC_r =4^n .

Prove that underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0 , where k = (3n)/2 and n is an even integer.

Evaluate : underset (r=1) overset (n) sum ""^nC_r 2^r .

The value of ""^50C_4+ underset (r=1) overset (6) sum ""^(56-r)C_3 is :

The value of C (47, 4) + underset (r=1) overset (5) sum (52-r,3) is:

Prove that underset(0)overset(1)intx(1-x)^5dx = 1/42

If x_1,x_2,………….x_p are p values of variabled x such that underset(i=1)overset(p)sum(x_i-2)=110 and underset(i=1)overset(p)sum(x_i-5)=20 find the value of p and the mean.

Prove that following: underset(1)overset(5)intx-3dx .

Prove that : underset(0)overset(2a)intf(x) dx = underset(0)overset(2a)intf(2a-x)dx

Prove that following: underset(2)overset(5)int|x-1|dx