Question

What is the coeffecient of x^9 in the product (x-21C0) (X-21C1) … (X-21C10)?

Answer 1

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Step-by-step explanation:

Let us consider a simpler case (x−a)(x−b)(x−c)(x−d)(x−e) .

Maximum power of the expression is x5

The coefficient of x4 : (one less than the maximum power of x) = −(a+b+c+d+e)

⟹: sum of all the individual constants * (-1)

The coefficient of x3 : (two less than the maximum power of x) = (ab+ac+ad+ae+bc+bd+be+cd+ce+de)

⟹ Sum of product of 2 constants at a time

Extending this logic for the given expression:

Coefficient of x9 = Sum of product of 2 constants at a time

=(210)∗[(211)+(212)+…+(2110)]+

(211)∗[(212)+(213)+…+(2110)]+

(212)∗[(213)+(214)+…+(2110)]+

(213)∗[(214)+(215)+…+(2110)]+

(214)∗[(215)+(216)+…+(2110)]+

(215)∗[(216)+(217)+…+(2110)]+

(216)∗[(217)+(218)+…+(2110)]+

(217)∗[(218)+(219)+(2110)]+

(218)∗[(219)+(2110)]+

(219)∗[(2110)]