Question

A variable takes value x with frequency ${}^{n+x-1}{C}_x,x$ = 0, 1, 2, . . . n. The mode of the variable is ___.

Answer 1

Frequency for variable x is ${}^{n+x-1}{C}_x$
Where x = 0, 1, 2, . . . . . n.
Mode is the variable for which freq. is max.

The frequencies of the observations are ${}^{n-1}{C}_0{,}^n{C}_1{,}^{n+1}{C}_2\dots {}^{2n-1}{C}_n$.

Two corresponding frequencies in the sequence are of the form

${}^n{C}_r$ and ${}^{n+1}{C}_{r+1}$.

Since ${}^{n+1}{C}_{r+1}={\frac{n+1}{r+1}}^n{C}_r$.

${}^{n+1}{C}_{r+1}{>}^n{C}_r$ $[\therefore n\ge r]$

So, the observation with highest frequency is ${}^{2n-1}{C}_n$.

Therefore the mode is $n$