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# Solve the following

Question:

If nP4 = 24. nC5, then the value of n is ____________.

Solution:

Given :- ${ }^{n} P_{4}=24{ }^{n} C_{5}$

Since ${ }^{n} C_{5}=\frac{{ }^{n} P_{5}}{5 !} \quad\left(\because{ }^{n} C_{r}=\frac{{ }^{n} P_{r}}{r !}\right)$

$\therefore{ }^{n} P_{4}=\frac{24{ }^{n} P_{5}}{5 !}$

$\therefore \frac{{ }^{n} P_{5}}{{ }^{n} P_{4}}=\frac{5 !}{24}=\frac{5 \times 4 \times 3 \times 2}{24}$

i. e $\frac{n !}{(n-5) !} \times \frac{(n-4) !}{n !}=5$

i.e n − 4 = 5

i.e n = 9