Question

If $2 \times {}^n{C_5} = 9 \times {}^{n - 2}{C_5},$ then the value of n will be?
$
  1)7 \\
  2)10 \\
  3)9 \\
  4)5 \\
 $

Answer 1

Hint: Combination can be represented as the selection of the items where the order of the items does not matter and it gives the number of ways of the selection of the items and is expressed as ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$here we will expand the given expression by using the formula and simplify to get the required value for “n”.

Complete step-by-step answer:
Take the given expression: $2 \times {}^n{C_5} = 9 \times {}^{n - 2}{C_5}$
Expand the above expression by using the standard formula, ${}^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$
$2 \times \dfrac{{n!}}{{5! \times (n - 5)!}} = 9 \times \dfrac{{(n - 2)!}}{{(n - 2 - 5)!5!}}$
Simplify the above expression, and always remember that when you combine two negative terms, you have to add the terms and give negative sign to the resultant value.
$2 \times \dfrac{{n!}}{{5! \times (n - 5)!}} = 9 \times \dfrac{{(n - 2)!}}{{(n - 7)!5!}}$
Expand the factorial of the above term –
\[2 \times \dfrac{{n(n - 1)(n - 2)!}}{{5! \times (n - 5)(n - 6)(n - 7)!}} = 9 \times \dfrac{{(n - 2)!}}{{(n - 7)!5!}}\]
Like terms from both the sides of the equation cancels each other.
\[2 \times \dfrac{{n(n - 1)}}{{(n - 5)(n - 6)}} = 9\]
Multiply the brackets and simplify the expression –
\[\dfrac{{2{n^2} - 2n}}{{{n^2} - 11n + 30}} = 9\]
Cross multiply the above expression, where the denominator of one side is multiplied with the numerator of the opposite side.
\[
  2{n^2} - 2n = 9({n^2} - 11n + 30) \\
  2{n^2} - 2n = 9{n^2} - 99n + 270 \;
 \]
Move all the terms on one side of the equation then the sign of the terms also changes. Positive term becomes negative and vice-versa.
\[0 = 9{n^2} - 99n + 270 - 2{n^2} + 2n\]
Combine like terms together and simplify –
\[\underline {9{n^2} - 2{n^2}} + \underline {2n - 99n} + 270 = 0\]
Simplify –
\[7{n^2} - 97n + 270 = 0\]
Split the middle term –
\[7{n^2} - 70n - 27n + 270 = 0\]
Make pair of first two and last two terms –
\[\underline {7{n^2} - 70n} - 2\underline {7n + 27} 0 = 0\]
Take common multiple –
$
  7n(n - 10) - 27(n - 10) = 0 \\
  (n - 10)(7n - 27) = 0 \;
 $
$
  n - 10 = 0 \\
  n = 10 \\
 $
or

$
  7n - 27 = 0 \\
  7n = 27 \\
  n = \dfrac{{27}}{7} \;
 $
Hence, from the given multiple choices -the second option is the correct answer.
So, the correct answer is “Option B”.

Note: Always remember and don’t get confused between the permutations and combinations. Permutations are defined as the number of ways of the selection and the arrangement of the items in which order of the item is important and it is stated as \[{}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}\]. Also remember the standard formulas and the difference between the two and be good in factorial and the multiples of the number.