The difference between the maximum values of \({ }^6 \mathrm{C}_{\mathrm{r}}\) and \({ }^n \mathrm{C}_{\mathrm{r}} 16\), then \(\mathrm{n}=\)

The maximum value of \({ }^6 \mathrm{C}_{\mathrm{r}}\) occurs at \(\mathrm{r}=\frac{6}{2}=3\)
\(\therefore{ }^6 \mathrm{C}_3=\frac{6 !}{3 ! 3 !}=\frac{6 \times 5 \times 4}{6}=20\)
As per data given \(\left|{ }^6 \mathrm{C}_3-{ }^{\mathrm{n}} \mathrm{C}_3\right|=16\)
\(\therefore{ }^{\mathrm{n}} \mathrm{C}_3=20+16 \text { or }{ }^{\mathrm{n}} \mathrm{C}_3=20-16\)
If \({ }^n C_3=36 \Rightarrow \frac{n !}{(n-3) ! n !}=36 \Rightarrow n(n-1)(n-2)=216\) is not possible for
\(\mathrm{n} \in \mathrm{N}\)
\(\text {If }{ }^{\mathrm{n}} \mathrm{C}_3=4 \Rightarrow \frac{\mathrm{n} !}{(\mathrm{n}-3) ! 3 !}=4 \Rightarrow \mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)=24 \)
\(\Rightarrow \mathrm{n}=4\)