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3 marksThe Binomial Theorem

CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2 · Question 5(a)(iii)

\binom{n}{r} is defined as the number of ways of selecting r distinct objects from a given set of n distinct objects.

Hence, prove that \left[ \binom{8}{6} + \binom{8}{5} \right] \times \left[ \binom{8}{3} + \binom{8}{2} \right] is a perfect square.

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Other parts of this question

  1. 5(a)(i)From the definition, show that \binom{n}{r} = \binom{n}{n - r}.[2 marks]
  2. 5(a)(ii)From the definition, show that \binom{n+1}{r} = \binom{n}{r} + \binom{n}{r - 1}.[4 marks]
  3. 5(b)(i)Find the number of 5-digit numbers greater than 30 000 which can be formed with the digits, 1, 3, 5, 6 and 8, if no digit is repeated.[3 marks]
  4. 5(b)(ii)What is the probability of one of the numbers chosen in (b)(i) being even?[5 marks]
  5. 5(c)(i) a)Show that (1 - i) is one of the square roots of -2i.[2 marks]
  6. 5(c)(i) b)Find the other square root of -2i.[1 mark]
  7. 5(c)(ii)Hence, find the roots of the quadratic equation z^2 - (3 + 5i)z + (8i - 4) = 0.[5 marks]

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