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Question: How do you simplify \( ^7{P_5} \) ?....

How do you simplify 7P5^7{P_5} ?.

Explanation

Solution

In this question we need to simplify 7P5^7{P_5} . Here, we will use the formula for permutations to simplify this. The formula of permutation is nPr=n!(n−r)!^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}} . This refers to the arrangement of all the members of a set in some order or sequence.

Complete step-by-step solution:
Here, we need to simplify 7P5^7{P_5} .
7P5^7{P_5} is in the form of nPr^n{P_r} , therefore we can use the formula nPr=n!(n−r)!^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}} .
Here nPr^n{P_r} represents nn permutation rr .
Where nn is a set of things, and rr is the arrangement of things where 0<r⩽n0 < r \leqslant n .
The n!n! means the product of all positive integers less than or equal to nn .
We can see from the given term 7P5^7{P_5} in the place of nn we have 77 , which shows that n=7n = 7
Similarly, we can see that in the place of, we have 55 , which shows that r=5r = 5 .
Therefore, substituting the value of n=7n = 7 and r=5r = 5 in the formula of the permutations, we have,
nPr=n!(n−r)!^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}
7P5=7!(7−5)!^7{P_5} = \dfrac{{7!}}{{\left( {7 - 5} \right)!}}
7P5=7!2!^7{P_5} = \dfrac{{7!}}{{2!}}
7P5=7×6×5×4×3×2×12×1^7{P_5} = \dfrac{{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}}
7P5=7×6×5×4×3^7{P_5} = 7 \times 6 \times 5 \times 4 \times 3
7P5=2520^7{P_5} = 2520

Hence, the permutation of 7P5^7{P_5} is 25202520 .

Note: It is important to note here that permutation refers to the process of arranging all the members of a given set to form a sequence without replacement. The number of permutations on a set of nn elements is given by n!n! , where ‘ !! ’ represents factorials. When we come across the term permutation, we usually here, about the term combination. A combination is the choice of rr things from a set of nn things without replacement. The order does not matter in combination. It is given by the formula nCr=n!r!(n−r)!^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}} . Permutations and combinations help us to determine the number of different ways of arranging and selecting objects without actually listening to them in real life.