If ⁹P₅ + 5. ⁹P₄ = ¹⁰P_r, then the value of r is

Concept:

PERMUTATION:

Number of Permutations of ‘n’ things taken ‘r’ at a time:

\({\rm{P}}\left( {{\rm{n}},{\rm{\;r}}} \right) ={}^nP_r =\frac{{{\rm{n}}!}}{{\left( {{\rm{n}} - {\rm{r}}} \right)!}}\)

Calculation:

\({}_{}^9{P_5} + 5.{}_{}^9{P_4} = {}_{}^{10}{P_r}\)

\(\frac{{9!}}{{4!}} + 5.\frac{{9!}}{{5!}} = \frac{{10 × 9!}}{{\left( {10 - r} \right)!}}\)

\(9![\frac{{1}}{{4!}} + \frac{{5}}{{5!}}] = \frac{{10 × 9!}}{{\left( {10 - r} \right)!}}\)

\([\frac{{1}}{{4!}} + \frac{{5}}{{5×4!}}] = \frac{{10 }}{{\left( {10 - r} \right)!}}\)

\( \Rightarrow \frac{2}{{4!}} = \frac{{10}}{{\left( {10 - r} \right)!}}\)

\( \Rightarrow \frac{1}{{4!}} = \frac{{5}}{{\left( {10 - r} \right)!}}\)

(10 – r)! = 5 × 4!

(10 – r)! = 5!

r = 5