Question

a senate committee has $5$ republicans and $4$ democrats. in how many ways can the committee members sit in a row of $9$ chairs, such that all $4$ democrats sit together?

Answer 1

There are 6! ways to arrange the 5 Republicans and 1 Democrat block, and 4! ways to rearrange the Democrats within their block, totaling 17,280 arrangements.

Step-by-step explanation:

To calculate the number of ways a senate committee consisting of 5 Republicans and 4 Democrats can sit in a row such that all 4 Democrats sit together, we can treat the group of Democrats as a single unit initially. This approach gives us 6 units to arrange: 5 individual Republicans and 1 block of Democrats. As there are 6 units, there are 6! (factorial) ways to arrange them.

However, the 4 Democrats can also be rearranged among themselves within their block in 4! ways. Thus, for each arrangement of the 6 units, there are 4! ways to arrange the Democrats. The total number of arrangements is the product of these two quantities, which equals to 6! × 4!.

Now, calculating these factorials, we get:

• 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
• 4! = 4 × 3 × 2 × 1 = 24

Therefore, the total number of ways the committee members can be arranged is 720 × 24 = 17,280.