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Considering the definition of permutation, in 5040 ways can the letters in the word ''PAYMENT'' be arranged if the letters are taken 6 at a time.

What is Permutation

Permutation is placing elements in different positions. So, permutations of m elements in n positions are called the different ways in which the m elements can be arranged occupying only the n positions.

That is, permutations refer to the action of arranging all the members of a set in some sort of order or sequence.

In other words, permutations (or Permutations without repetition) are ways of grouping elements of a set in which:

  • take all the elements of a set.
  • the elements of the set are not repeated.
  • order matters.

To obtain the total of ways in which m elements can be placed in n positions, the following expression is used:

[tex]mPn=\frac{m!}{(m-n)!}[/tex]

where "!" indicates the factorial of a positive integer, which is defined as the product of all natural numbers before or equal to it.

This case

In this case, you have:

  • the letter word "PAYMENT", where the number of letters is 7.
  • the letters are taken 6 at a time.

Then, you know that:

  • m= 7
  • n=6

Replacing in the definition of permutation:

[tex]7P6=\frac{7!}{(7-6)!}[/tex]

Solving:

[tex]7P6=\frac{7!}{1!}[/tex]

7P6= 5040

Finally, in 5040 ways can the letters in the word ''PAYMENT'' be arranged if the letters are taken 6 at a time.

Learn more about permutation:

brainly.com/question/12296796

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