Gordon buys 5 dolls for his 5 nieces. The gifts include 2 identical Sun - and Fun beach dolls, 1 Elegant Eddie dress - up doll 1 G.I. Joise army doll, and 1 Tulip Troll doll. If the youngest niece does not want the G.I. Josie doll, in how many different ways can he give the gifts?

Answer

To solve the problem of how many different ways Gordon can give the dolls to his five nieces, we need to consider the restrictions given in the problem. Here’s a step-by-step solution:

### Step 1: Identify the Dolls and Niece Preferences
Gordon has 5 dolls:
- 2 identical Sun and Fun beach dolls (let's denote them as S, S)
- 1 Elegant Eddie dress-up doll (E)
- 1 G.I. Josie army doll (G)
- 1 Tulip Troll doll (T)

The youngest niece does not want the G.I. Josie doll (G).

### Step 2: Determine the Possible Assignments
Since the youngest niece does not want the G.I. Josie doll, she can only receive one of the other dolls: either E, S, or T.

### Step 3: Calculate the Assignments for Each Case
1. **Case 1: Youngest niece receives E**
- Remaining dolls: S, S, G, T
- The number of ways to assign these 4 dolls is calculated using the formula for permutations of multiset:
\[ \text{Ways} = \frac{4!}{2!} = \frac{24}{2} = 12 \]

2. **Case 2: Youngest niece receives T**
- Remaining dolls: S, S, E, G
- The number of ways to assign these 4 dolls is:
\[ \text{Ways} = \frac{4!}{2!} = \frac{24}{2} = 12 \]

3. **Case 3: Youngest niece receives S (one of the S dolls)**
- Remaining dolls: S, E, G, T
- The number of ways to assign these 4 dolls is:
\[ \text{Ways} = 4! = 24 \]

### Step 4: Total the Ways
Now, we can sum the ways from all three cases:
\[ \text{Total Ways} = 12 + 12 + 24 = 48 \]

### Conclusion
Thus, the total number of different ways Gordon can give the gifts to his nieces is **48**.

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