Question

If n is even and the value of ${}^n{C_r}$ is maximum, then $r = $
$
  1)\dfrac{n}{2} \\
  2)\dfrac{{n + 1}}{2} \\
  3)\dfrac{{n - 1}}{2} \\
$
$4)$ None of these

Answer 1

Hint: Combination can be well-defined 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 find the maximum value by using the binomial coefficient series.

Complete step-by-step answer:
We know that the binomial coefficient series can be given as –
${}^n{C_0},{}^n{C_1},{}^n{C_2},.....{}^n{C_{r,}}{}^n{C_{n - 2}},{}^n{C_{n - 1}},{}^n{C_n}$
Total number of terms can be given as: $n + 1,r = 0\;to{\text{ n}}$, since we have to find the maximum value of ${}^n{C_r}$
By comparison, ${}^n{C_r}$the middle term will be $(r + 1)$
Middle term $ = \dfrac{{(n + 1 + 1)}}{2}$
Simplify the above expression –
$r + 1 = \dfrac{{(n + 2)}}{2}$
Make the required term the subject and move other terms on the opposite side. When you move any term from the left hand side to the right hand side of the equation, the sign of the term also changes. Positive becomes negative and vice-versa.
$r = \dfrac{{(n + 2)}}{2} - 1$
Simplify the above equation finding the LCM (least common factor) of the above expression –
$r = \dfrac{{(n + 2) - 2}}{2}$
Simplify the above expression –
$r = \dfrac{{n + 2 - 2}}{2}$
Like terms with the same value and the opposite sign cancels each other.
$r = \dfrac{n}{2}$
Hence, from the given multiple choices – the first option is the correct answer.
So, the correct answer is “Option 1”.

Note: Don’t get confused between the permutations and combinations. Permutations can be 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 expressed as \[{}^n{P_r} = \dfrac{{n!}}{{(n - r)!}}\]. Also remember the standard formulas and the difference between the two.