Question:
If ${ }^{n} P_{r}=840$ and ${ }^{n} C_{r}=35$, find the value of $r$.
Solution:
Given: ${ }^{n} P_{r}=840$ and ${ }^{n} C_{r}=35$
To find: $r=?$
We know that:
${ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=\frac{n !}{(n-r) ! \times r !}$
and
$\mathrm{n} \mathrm{P}_{\mathrm{r}}=\frac{n !}{(n-r) !}$
$\Rightarrow{ }^{n} P_{r}={ }^{n} C_{r} \times r !$
$\Rightarrow 840=35 \times r !$
⇒ $r !=\frac{840}{35}=24$
$\Rightarrow r !=4 !$
$\Rightarrow r=4$
Ans: $r=4$
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