Consider $$n$$ points in a plane no three of which are collinear and the ratio of number of hexagon and octagon that can be formed from these $$n$$ points is $$4 : 13$$ then find the value of $$n.$$
A.
14
B.
20
C.
28
D.
None of these
Answer :
20
Solution :
From $$n$$ points number of hexagon is $$^n{C_6} = \frac{{n!}}{{\left\{ {\left( {n - 6} \right)!} \right\}\left\{ {6!} \right\}}}$$
From $$n$$ points number of Octagons is $$^n{C_8} = \frac{{n!}}{{\left\{ {\left( {n - 8} \right)!} \right\}\left\{ {8!} \right\}}}$$
Ratio of number of hexagon to number of octagon is
$$\frac{{\left\{ {\left( {n - 8} \right)!} \right\}\left\{ {8!} \right\}}}{{\left\{ {\left( {n - 6} \right)!} \right\}\left\{ {6!} \right\}}} = \frac{{7 \times 8}}{{\left( {n - 7} \right)\left( {n - 6} \right)}} = \frac{4}{{13}}$$
On solving this quadratic equation we will get $$n = 20$$
Releted MCQ Question on Algebra >> Permutation and Combination
Releted Question 1
$$^n{C_{r - 1}} = 36,{\,^n}{C_r} = 84$$ and $$^n{C_{r + 1}} = 126,$$ then $$r$$ is:
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are
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