Seating Arrangements for Men and Women

This problem asks us to determine the number of ways to seat 5 women and 3 men in a row such that two men are always together. To solve this, we will treat the two men who must sit together as a single unit or a block. The key is to correctly identify the entities we are arranging and the internal arrangements within any grouped units.

Step-by-Step Seating Arrangement Calculation

Let's break down the process into clear steps:

1. Forming the "Two Men Together" Group:
   • First, we need to choose which 2 out of the 3 men will form the group that sits together. The number of ways to choose 2 men from 3 is given by the combination formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). So, \(\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{(2 \times 1)(1)} = 3\) ways.
   • Once these 2 men are chosen, they can be arranged among themselves within their group. Since there are 2 men, they can be arranged in \(2!\) ways. So, \(2! = 2 \times 1 = 2\) ways.
   • Therefore, the total number of ways to form this specific "two men together" block (let's call it 'B') is the product of the ways to choose them and the ways to arrange them internally: Number of ways to form block 'B' = \(\binom{3}{2} \times 2! = 3 \times 2 = 6\) ways.
2. Identifying the Entities for Arrangement:
   • After forming the block 'B' (containing 2 men), we are left with:
     • The block 'B' itself (acting as one unit).
     • The one man who was not chosen for the block (let's call him \(M_{single}\)).
     • The 5 women (\(W_1, W_2, W_3, W_4, W_5\)).
   • In total, we now have \(1 (\text{block B}) + 1 (M_{single}) + 5 (\text{women}) = 7\) distinct entities to arrange in a row.
3. Arranging the Entities in a Row:
   • The number of ways to arrange \(n\) distinct entities in a row is given by \(n!\). In our case, we need to arrange 7 distinct entities. So, the number of ways to arrange these 7 entities is \(7!\).
   • Calculating \(7!\): \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\) ways.
4. Calculating the Total Number of Ways:
   • To find the final total number of ways, we multiply the number of ways to form the "two men together" block by the number of ways to arrange all the resulting entities.
   • Total ways = (Ways to form block 'B') \(\times\) (Ways to arrange 7 entities)
   • Total ways = \(6 \times 5040 = 30240\).

Summary of Seating Possibilities

The problem is a classic example of permutations with a grouping constraint. By treating the two men who must sit together as a single unit, we simplify the problem into arranging fewer, larger entities, and then account for the internal arrangements within the grouped unit.

Thus, there are 30240 ways to seat 5 women and 3 men in a row such that two men are together.