Question

Find the value of ‘r’ for which.
\[{20_{{{\text{C}}_{\text{r}}}}}{20_{{{\text{C}}_{\text{o}}}}} + {20_{{{\text{C}}_{{\text{r}} - 1}}}}{20_{{{\text{C}}_1}}} + {20_{{{\text{C}}_{{\text{r}} - 2}}}}{20_{{{\text{C}}_2}}} + \ldots + {20_{{{\text{C}}_{\text{o}}}}}{20_{{{\text{C}}_{\text{r}}}}}\] is maximum.
A) 20
B) 10
C) 15
D) 11

Answer 1

Hint: Here, we will be using the Vandermonde’s formula to solve the question’s first part i.e. first we will find the value of the expression and then we will find the value of ‘r’ for which the expression attains a maximum.

Complete step by step solution: Formula used: Vandermonde’s Formula:
This is the direct formula which we can apply to solve the problem. It says:-
\[{{\text{n}}_{{{\text{C}}_{\text{r}}}}}{{\text{n}}_{{{\text{C}}_{\text{o}}}}} + {{\text{n}}_{{{\text{C}}_{{\text{r}} - 1}}}}{{\text{n}}_{{{\text{C}}_1}}} + {{\text{n}}_{{{\text{C}}_{{\text{r}} - 2}}}}{{\text{n}}_{{{\text{C}}_2}}} + \ldots + {{\text{n}}_{{{\text{C}}_{\text{o}}}}}{{\text{n}}_{{{\text{C}}_{\text{r}}}}} = 2{{\text{n}}_{{{\text{C}}_{\text{r}}}}}\]
Here one most remember following points:-
a) \[{{\text{n}}_{{{\text{C}}_{\text{r}}}}}{{\text{n}}_{{{\text{C}}_{\text{o}}}}} + {{\text{n}}_{{{\text{C}}_{{\text{r}} - 1}}}}{{\text{n}}_{{{\text{C}}_1}}} + {{\text{n}}_{{{\text{C}}_{{\text{r}} - 2}}}}{{\text{n}}_{{{\text{C}}_2}}} + \ldots + {{\text{n}}_{{{\text{C}}_{\text{o}}}}}{{\text{n}}_{{{\text{C}}_{\text{r}}}}} = 2{{\text{n}}_{{{\text{C}}_{\text{r}}}}}\] n is same in all the terms
b) base terms should add upto same number
here r + o = r | r − 1 + 1 = r | r − 2 + 2 = r | … o + r |
c) and the sum of expression is $2 \times {{\text{n}}_{{{\text{C}}_{\text{r}}}}}$ i.e
2 × power Csum of bases.
So, if we substitute the expression in the Vandermonde’s formula, we get
\[{20_{{{\text{C}}_{\text{r}}}}}{20_{{{\text{C}}_{\text{o}}}}} + {20_{{{\text{C}}_{{\text{r}} - 1}}}}{20_{{{\text{C}}_1}}} + {20_{{{\text{C}}_{{\text{r}} - 2}}}}{20_{{{\text{C}}_2}}} + \ldots + {20_{{{\text{C}}_{\text{o}}}}}{20_{{{\text{C}}_{\text{r}}}}}\]
= $2 \times {20_{{{\text{C}}_{\text{r}}}}}$
= ${40_{{{\text{C}}_{\text{r}}}}}$
Now ${40_{{{\text{C}}_{\text{r}}}}}$ attains maximum at ${\text{r}} = \dfrac{{40}}{2} = 20$.
(Note:- ${n_{{{\text{C}}_{\text{r}}}}}$ attains maximum at ${\text{r}} = \dfrac{{\text{n}}}{2}$ if n is even
and ${\text{r}} = \dfrac{{{\text{n}} + 1}}{2}$ if n is odd)

∴ Correct answer is (A) 20.

Note: One must be very careful in applying the Vandermonde’s formula as one should also check that it contains all the terms i.e from \[{{\text{n}}_{{{\text{C}}_{\text{o}}}}}\] \[{{\text{n}}_{{{\text{C}}_{\text{r}}}}}\] … \[{{\text{n}}_{{{\text{C}}_{\text{r}}}}}\] \[{{\text{n}}_{{{\text{C}}_{\text{o}}}}}\].
If any one of the term is even missing, you cannot apply the formula.