Question

in how many ways can the letters in the word payment be arranged if the letters are taken 6 at a time

Answer 1

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:

$mPn=\frac{m!}{(m-n)!}$

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:

$7P6=\frac{7!}{(7-6)!}$

Solving:

$7P6=\frac{7!}{1!}$

7P6= 5040

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

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