Recall the formulas for the number of permutations and combinations of n objects taken r at a time, where r ≤ n.
Equation: _n C_r * r! = _nP_r Value: 24 Interpretation:There are 24 times more permutations than combinations.
Practice makes perfect
We will start by writing an equation that relates _nP_r and _nC_r. Let's begin by recalling the formulas for the number of permutations and combinations of n objects taken r at a time, where r ≤ n.
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Permutations & & Combinations [0.6em]
_n P_r = n!/(n-r)! & & _n C_r = n!/(n-r)! * r!
The numerators of both formulas are the same and equal to n!. We will analyze the denominators of the above formulas. Notice the expression (n-r)! is common for both denominators. However, in the formula for combinations there is an extra r!.
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Permutations & & Combinations [0.6em]
_n P_r = n!/(n-r)! & & _n C_r = n!/(n-r)! * r!Therefore, in order write an equation relating _nP_r and _nC_r we start by rewriting the formula for _n C_r as a product of fractions.
We found the equation that relates _nP_r and _nC_r. We can use it to find the value of the given quotient. Let's substitute 182 for n and 4 for r in the obtained relation.
_n C_r * r! = _nP_r
⇓
_(182) C_4 * 4! = _(182) P_4
We are ready to substitute it and evaluate the given quotient.
The quotient can be used to compare the number of permutations and combinations. If we divide the number of permutations by the number of combinations, we know how many times more permutations than combinations there are.
Therefore, we know there are 24 times more permutations than combinations of 182 objects taken 4 at a time.
