Using nCr=n!r!(n−r)!{}^nC_r = \frac{n!}{r!(n-r)!}nCr=r!(n−r)!n!, show that nC4=n(n−1)(n−2)(n−3)24{}^nC_4 = \frac{n(n-1)(n-2)(n-3)}{24}nC4=24n(n−1)(n−2)(n−3).
A researcher is selecting distinct plant species from a population of nnn available species for a DNA sequencing study. Show that the equation 2×nC4=15×nC22 \times {}^nC_4 = 15 \times {}^nC_22×nC4=15×nC2 simplifies to n2−5n−84=0n^2 - 5n - 84 = 0n2−5n−84=0
Hence, solve the equation 2×nC4=15×nC22 \times {}^nC_4 = 15 \times {}^nC_22×nC4=15×nC2