Question

# Find the value of n and r, if $^n\Pr$=720 and $^nCr$=120

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Hint: A permutation is defined as an arrangement in a definite order of a number of objects taken some or all at a time. The convenient expression to denote permutation is defined as
The permutation formula is given by,
$^n\Pr = \dfrac{{n!}}{{\left( {n - r} \right)!}};0 \leqslant r \leqslant n$
Where the symbol denotes the factorial which means that the product of all the integer less than or equal to n but it should be greater than or equal to 1.
Combination- the combination is a selection of a part of a set of objects or selection of all objects when the order does not matter. Therefore, the number of combinations of n objects taken r at a time and the combination formula is given by,
$^nCr = \dfrac{{n(n - 1)(n - 2).....(n - r + 1)}}{{r!}}$
$= \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
$^nCr = \dfrac{{^n\Pr }}{{r!}}$
Therefore,

Given, $^n\Pr = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
Which is equal to $720$
$\dfrac{{n!}}{{\left( {n - r} \right)!}} = 720$
As we know the relation between permutation and combination
$^n\Pr = r{!^n}Cr$
$\dfrac{{^n\Pr }}{{^nCr}} = r!.......1.$
Also given in the question $^nCr = 120$
Putting the value of $^n\Pr$ and $^nCr$ in equation 1.
$\dfrac{{720}}{{120}} = r!$
$r! = 6$
$r! = 3 \times 2 \times 1$
$r = 3$
Now, $^n\operatorname{P} 3 = 720$
We can write this
$n(n - 1)(n - 2) = 720$
$n(n - 1)(n - 2) = 10 \times 9 \times 8$
From this we get
$n = 10$
Hence the value of $n$ and $r$ are $10$ and $3$ respectively.

Note: The relation between permutation and combination-
$^n\Pr { = ^n}Cr.r!$ if
$0 < r \leqslant n$
$^nCr{ + ^n}Cr - 1{ = ^{n + 1}}Cr$
The fundamental principle of counting
Multiplication principal
Suppose an operation
The fundamental principle of counting –
Multiplication principle: suppose an operation A can be performed in m ways and associated with each way of performing another operation B can be performed in n ways, then the total number of performances of two operations in the given order is $m \times n$ ways. This can be extended to any finite number of operations.
Addition principle: if an operation A can be performed in m ways and another operation S, which is independent of A, can be performed in $m + n$ ways. This can be extended to any finite number of exclusive events.