Find the differences between the largest values in the following:… - Vidyadip

Question

Find the differences between the largest values in the following: ${ }^{13} \mathrm{C}_{\mathrm{r}}-{ }^8 \mathrm{C}_{\mathrm{r}}$

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Answer

Greatest value of ${ }^{13} \mathrm{C}_{\mathrm{r}}$
Here $\mathrm{n}=13$, which is odd
Greatest value of ${ }^n C_r$ occurs at $r=\frac{n-1}{2}$ if $n$ is odd
$
\begin{array}{ll}
\therefore & r=\frac{n-1}{2} \\
\therefore & r=\frac{13-1}{2}=6
\end{array}
$
$
\begin{aligned}
& \therefore \quad \text { Greatest value of }{ }^{13} \mathrm{C}_{\mathrm{r}}={ }^{13} \mathrm{C}_6=\frac{13 !}{6 ! 7 !} \\
&=\frac{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 7 !} \\
&=1716
\end{aligned}
$
Also, for greatest value of ${ }^8 \mathrm{C}_{\mathrm{r}}$ $\mathrm{n}=8$, which is even
$
\begin{aligned}
\therefore \quad r= & \frac{8}{2}=4 \\
\therefore \quad{ }^8 C_4 & =\frac{8 !}{4 ! 4 !} \\
& =\frac{8 \times 7 \times 6 \times 5 \times 4 !}{4 \times 3 \times 2 \times 1 \times 4 !} \\
& =70
\end{aligned}
$
$\therefore$ Difference between the greatest values of ${ }^{13} \mathrm{C}_{\mathrm{r}}$ and ${ }^8 \mathrm{C}_{\mathrm{r}}=1716-70=1646$

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