Prove that , `.^(15)C_(8)+^(15)C_(9)-.^(15)C_(6)-^(15)C_(7)=0`

""^(15)C_(3) + ""^(15)C_(5) + ……… +""^(15)C_(15) =

The value of .^(15)C_(0)^(2)-.^(15)C_(1)^(2)+.^(15)C_(2)^(2)-"...."-.^(15)C_(15)^(2) is a. 15 b. -15 c. 0 d. 51

Given are six 0' s, five 1' s and four 2's . Consider all possible permutations of all these numbers. [A permutations can have its leading digit 0 ]. How many permutations have the first 0 preceding the first 1 ? (a) "^(15)C_(4)xx^(10)C_(5) (b) "^(15)C_(5)xx^(10)C_(4) (c) "^(15)C_(6)xx^(10)C_(5) (d) "^(15)C_(5)xx^(10)C_(5)

If |{:(.^(9)C_(4),.^(9)C_(5),.^(10)C_(r)),(.^(10)C_(6 ),.^(10)C_(7),.^(11)C_(r+2)),(.^(11)C_(8),.^(11)C_(9),.^(12)C_(r+4)):}|=0 , then the value of r is equal to

Find the following sums : (i) .^(n)C_(0)-.^(n)C_(2)+.^(n)C_(4)-.^(n)C_(6)+"....." (ii) .^(n)C_(1)-.^(n)C_(3)+.^(n)C_(5)-.^(n)C_(7)+"...." (iii) .^(n)C_(0)+.^(n)C_(4)+.^(n)C_(8)+.^(n)C_(12)+"....." (iv) .^(n)C_(2) + .^(n)C_(6) + .^(n)C_(10)+.^(n)C_(14)+"......" (v) .^(n)C_(1) + .^(n)C_(5)+.^(n)C_(9)+.^(n)C_(13)+"...." (vi) .^(n)C_(3) + .^(n)C_(7) + .^(n)C_(11) + .^(n)C_(15) + "....."

The value of .^(20)C_(0)+.^(20)C_(1)+.^(20)C_(2)+.^(20)C_(3)+.^(20)C_(4)+.^(20)C_(12)+.^(20)C_(13)+.^(20)C_(14)+.^(20)C_(15) is

If a= .^(20)C_(0) + .^(20)C_(3) + .^(20)C_(6) + .^(20)C_(9) + "…..", b = .^(20)C_(1) + .^(20)C_(4) + .^(20)C_(7) + "……"' and c = .^(20)C_(2) + .^(20)C_(5) + .^(20)C_(8) + "…..", then Value of a^(3) + b^(3) + c^(3) - 3abc is

If .^(15)C_(r+1)=^(15)C_(r+2) , find r.

Find the sum of the series .^15 C_0+^(15)C_1+^(15)C_2+...............+^(15)C_7 .

Find the sum of the series .^(84)C_(4)+6xx.^(84)C_(5)+15xx.^(84)C_(6)+20xx.^(84)C_(7)+15xx.^(84)C_(8) +6xx.^(84)C_(9)+.^(84)C_(10) .