Evaluate expression 6p5
The Correct Answer and Explanation is:
The expression ( 6P5 ) represents a permutation, which is a way to calculate the number of possible arrangements of items when the order in which the items are arranged matters. The formula for permutations is:
[
nP r = \frac{n!}{(n-r)!}
]
Where:
- ( n ) is the total number of items,
- ( r ) is the number of items to be selected, and
- ( n! ) (n factorial) is the product of all positive integers up to ( n ), i.e., ( n! = n \times (n-1) \times (n-2) \times \dots \times 1 ).
In this case:
- ( n = 6 ) (the total number of items),
- ( r = 5 ) (the number of items to be arranged).
Now, using the permutation formula:
[
6P5 = \frac{6!}{(6-5)!} = \frac{6!}{1!}
]
First, calculate the factorials:
- ( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 ),
- ( 1! = 1 ).
Substitute these values into the formula:
[
6P5 = \frac{720}{1} = 720
]
Thus, the value of ( 6P5 ) is ( 720 ).
Explanation:
The concept of permutations deals with the arrangement of a certain number of objects, where the order is important. In this case, we’re arranging 5 objects out of 6 available objects. The formula for permutations helps us compute how many different ways we can arrange a subset of items, considering the distinct order of placement.
The expression ( 6P5 ) can be interpreted as selecting 5 positions from 6 available options and then arranging the 5 selected positions. The order matters, so each arrangement counts as a unique permutation. The fact that ( n! ) represents all possible ways to arrange the total set and ( (n – r)! ) accounts for the unchosen positions, leaving us with the total number of distinct ordered selections.