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

Question

Find the differences between the largest values in the following: ${ }^{14} \mathrm{C}_{\mathrm{r}}-{ }^{12} \mathrm{C}_{\mathrm{r}}$

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Answer

Greatest value of ${ }^{14} \mathrm{C}_{\mathrm{r}}$
Here $n=14$, which is even
Greatest value of ${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$ occurs at $\mathrm{r}=\frac{n}{2}$ if $\mathrm{n}$ is even
$
\begin{array}{ll}
\therefore & r=\frac{n}{2} \\
\therefore & r=\frac{14}{2}=7
\end{array}
$
$
\begin{aligned}
& \therefore \quad \text { Greatest value of }{ }^{14} \mathrm{C}_{\mathrm{r}}={ }^{14} \mathrm{C}_7=\frac{14 !}{7 ! 7 !} \\
&=\frac{14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 !}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 7 !} \\
&=3432
\end{aligned}
$
Also, for greatest value of ${ }^{12} \mathrm{C}_{\mathrm{r}}$ $\mathrm{n}=12$, which is even
$
\begin{aligned}
\therefore \quad r=\frac{12}{2} & =6 \\
\therefore \quad{ }^{12} \mathrm{C}_6 & =\frac{12 !}{6 ! 6 !} \\
& =\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 6 !} \\
& =924
\end{aligned}
$
$\therefore$ Difference between the greatest values of ${ }^{14} C_r$ and ${ }^{12} C_r=3432-924=$ 2508
\mathrm{C}_{\mathrm{r}}$ $\mathrm{n}=12$, which is even
$
\begin{aligned}
\therefore \quad \mathrm{r}=\frac{12}{2} & =6 \\
\therefore \quad{ }^{12} \mathrm{C}_6 & =\frac{12 !}{6 ! 6 !} \\
& =\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 !}{6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 6 !} \\
& =924
\end{aligned}
$
$\therefore$ Difference between the greatest values of ${ }^{14} C_r$ and ${ }^{12} C_r=3432-924=$ 2508