# Find the coefficient of x^8 in (x-1)(x-2)(x-3)................(x-10).

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Find the coefficient of x8 in (x-1)(x-2)(x-3)................(x-10).

by (64.2k points)
edited

Solution:

We know

(x - a)(x - b) = x2 - (a + b)x + ab

similarly(x - a)(x - b)(x - c) = x3 - (a + b + c)x2 + (ab + bc + ac)x

so (x - 1)(x - 2)...(x - 10) = x10 – (1 + 2 + 3 + . . . . . . . + 10) x9 + (1*2 + 2*3 + 3*4 + . . . . . . . + 9*10 + 10*1) x8...

so coeff of x= 2 + 6 + 12 + 20 + 30 .... + 90 + 10 = Σ n(n +1) = Σn2 + Σn + 10

Σ is summation sign. n is 9.

Σn2 + Σn + 10= n(n+1)/2 +n(n+1)(2n+1)/6

= 9(10)/2 + 9(10)(19)/6 + 10

= 9 x 5 + 3 x 5 x 19 + 10

= 45 + 285 + 10

= 340

by (16 points)
+1
In the last second step....calculation is wrong..
9×5=45  and not 135
by (64.2k points)
+1
thanks for correcting me:)
It was typo error