 QUESTION

# If the value of $^{10}{{P}_{r}}=5040$, then the value of r is:[a] 2[b] 3[c] 4[d] None of these

Hint: Start from r = 0 and check if LHS = RHS. If not put r =1 and check again. Repeat the process till you get LHS = RHS or r>10. If r>10 then there is no such value of r. Alternatively Divide RHS by 10, then 9 and continue till RHS = 1 or if RHS is not a natural number. If RHS is not a natural number, then there is no solution. Otherwise, the solution is 10 - the last number with which you divided +1.

We know that $^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}=n\left( n-1 \right)\ldots \left( n-r+1 \right)$
Now, put n = 10 and r =0 , we get
LHS ${{=}^{10}}{{P}_{0}}=1$
Since LHS $\ne$ RHS, we continue
Put n = 10, r= 1
LHS ${{=}^{10}}{{P}_{1}}=10$
Since LHS $\ne$ RHS, we continue
Put n = 10, r= 2
LHS ${{=}^{10}}{{P}_{2}}=10\times 9=90$
Since LHS $\ne$ RHS, we continue
Put n = 10, r= 3
LHS ${{=}^{10}}{{P}_{3}}=10\times 9\times 8=720$
Since LHS $\ne$ RHS, we continue
Put n = 10, r= 4
LHS ${{=}^{10}}{{P}_{4}}=10\times 9\times 8\times 7=720=5040$
Here LHS = RHS.
Hence r =4.
Hence option [c] is correct.

Note: Alternate solution:
RHS = 5040.
Dividing RHS by 10, we get RHS = 504
Since RHS $\ne 1$, we continue
Dividing RHS by 9, we get RHS = 56
Since RHS $\ne 1$, we continue

Dividing RHS by 8, we get RHS =7.
Since RHS $\ne 1$, we continue

Dividing RHS by 7, we get RHS = 1.
Since RHS = 1, we stop.
Hence r = 10-7+1=4.
Hence option [c] is correct.