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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,

**Complete step by step answer:**

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.

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