TTMUA Lab
TMUA 2019 · Paper 1 · Question 3 of 20

TMUA 2019 Paper 1 Question 3

Sequences and series — Binomial coefficients · summing a column of Pascal's triangle. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1Sequences and seriesBinomial coefficients · summing a column of Pascal's triangle8 options
Find the coefficient of x in the expression (1+x)0 + (1+x)1 + (1+x)2 + (1+x)3 + ⋯ + (1+x)79 + (1+x)80
  1. A80
  2. B81
  3. C324
  4. D628
  5. E3240
  6. F3321
  7. G6480
  8. H6642
Show the answer and worked solution
answer · E
  1. A80
  2. B81
  3. C324
  4. D628
  5. E3240
  6. F3321
  7. G6480
  8. H6642
There is no need to expand anything: the coefficient of x in (1+x)n is (n1) = n. So the total coefficient of x is 0 + 1 + 2 + ⋯ + 80 = 80 × 812 = 3240. The tempting slip is to count 81 brackets and reach for 81 × 822; the first bracket contributes nothing, so the sum stops at 80.