Find the value of n and r, if \[^n\Pr \]=720 and \[^nCr\]=120
Answer
Verified
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.
×
Sorry!, This page is not available for now to bookmark.