The number of ways of arrangement of 10 different things taken 4 at a time in which one particular thing always occurs?

Question

Answer

Arrangements of 10 different things taken 4 at a time with one particular thing always occurring

First, we choose the one particular thing that must always occur. We have 10 different things to choose from, so there are 10 options for this.

Next, we need to choose the other 3 things that will be included in the arrangement. We have 9 remaining things to choose from (since we have already chosen one), so there are 9 options for the first thing we choose, 8 options for the second thing we choose, and 7 options for the third thing we choose.

Once we have chosen all 4 things, we need to arrange them in a specific order. There are 4 things to arrange, so there are 4 options for the first thing, 3 options for the second thing, 2 options for the third thing, and 1 option for the fourth thing.

Putting it all together:

To find the total number of arrangements, we need to multiply the number of options for each step:

10 * 9 * 8 * 7 * 4 * 3 * 2 * 1 = 60,480

Therefore, there are 60,480 ways to arrange 10 different things taken 4 at a time in which one particular thing always occurs.