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Martha invited 4 friends to go with her to the movies. Find the ways in which Martha can be sitted in the middle.

This question aims to find how Martha can sit in the middle seat when she goes to see a movie with her four friends.

Martha booked 5 seats for a movie, 4 for her friends and one for herself. They all can sit in 120 possible ways in those 5 seats considering one person per seat. According to the given condition, Martha is sitting in the middle seat which means the third seat out of the 5 seats she booked.

She can sit in other seats in many possible ways. The first seat has four possible chances, the second seat has three possible chances, and the third seat has only one possible chance as Martha is sitting in that seat. The fourth seat has only two possible chances and the last seat which is the fifth seat has only one chance.

This possible arrangement can be calculated by using factorial calculation. Factorial is a way of analyzing the possible ways in which an object can be arranged. We can fix an object and find how it can be arranged.

The product of all positive integers that are less than or equal to the given positive integer is called a factorial. It is represented by that positive integer with an exclamation mark at the end.

Expert Answer

We can find the possible ways in which Martha can sit in the middle seat by using the factorial approach:

Number of ways = $  4 \times 3 \times 1 \times 2 \times 1  $

Number of ways can be represented by an integer n:

\[   n  =   4   \times   3   \times   1   \times   2   \times   1    \]

\[   n  =  24  \]

Numerical Solution

There are 24 possible ways in which Martha can sit in the middle seat.

Example

Find the number of ways in which the red toy car among the other 5 toy cars can be placed in the third section of a shelf. There is a space for only one toy car per section.

There is a total of 6 sections on a shelf in which we have to place these cars. They all can be placed in 720 possible ways in those 6 sections considering one toy car per section. According to the given condition, a red toy car is the most expensive one that must be placed in the center which means the third shelf.

The red toy car must be placed in the third section in many possible ways. The first section of the shelf has five possible chances, the second section has four possible chances, and the third section has one possible chance as a red toy car is placed in that section. The fourth section has only three possible chances and the fifth section has two possible chances the last section which is the sixth section has only 1 chance.

\[   n  =   5  \times   4   \times   1   \times   3  \times  2 \times 1  \]

\[   n  =  120  \]

Image/Mathematical drawings are created in Geogebra. 

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