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# The value of  30 C 0 - 30 C 1 + 30 C 2 - 30 C 3 +...+30 C 10 is

2075 Points
2 months ago
Note that coeff of x^10 in Y= (30C0.x^0 – 30C1.x^1 + …. + 30C30.x^30)*(1+x+x^2+.......) will be the reqd sum.
Y= (1 – x)^30.(1/(1 – x)) (using BT and infinite GP sum)
or Y= (1 – x)^29
so, coeff of x^10 is 29C10.
Thus, sum= 29C10.
KINDLY APPROVE :))
Ram Kushwah
108 Points
2 months ago
We know that

(1+x)n = ⁿCx + ⁿC1x+ ⁿC2x2+ ⁿC3x3+.....................................+ ⁿCnxn

Putting x = -1 we get

(1-1)n = ⁿC₀(-1)⁰ + ⁿC1(-1)+ ⁿC2(-1)2+ⁿC3(-1)3+.........................+ ⁿCn(-1)n

(1-1)n = ⁿC₀ - ⁿC1+ ⁿC2 - ⁿC3+.....................................+(-1)n ⁿCn

Further on plugging n=30 we get

30C - 30C1+ 30C2 - 30C3+..................+(-1)30 x 30C30 =030

OR  30C - 30C1 + 30C30C3+..................+ 30C30 = 0