Question

Miriam is visiting five friends in her neighborhood, Louise, Mike, Nicole, Oscar, and Pascal. In each arrangement below, the first initial of each person’s name represents that person’s position as he or she is visited by Miriam.

Which shows all the outcomes for this event if she visits with Nicole first and Louise last?

NMOPL, NOMPL, NPMOL
NLMOP, NLOMP, NMLOP, NMOLP, NOMNP, NONMP
NMOPL, NMPOL, NOPML, NOMPL, NPMOL, NPOML
NLMOP, NMLOP, NOLMP

Answer 1

To find all possible arrangements where Nicole is first and Louise is last, we start with the requirement that Miriam visits Nicole (N) first and Louise (L) last:

1. **Positions**:
- Position 1: N (Nicole)
- Position 5: L (Louise)

2. **Remaining Friends**:
The friends that can be arranged between Nicole and Louise are Mike (M), Oscar (O), and Pascal (P).

3. **Arranging Remaining Friends**:
The remaining positions (2, 3, and 4) can be filled with M, O, and P. The number of ways to arrange three items is calculated using the factorial of the number of items, which is 3! (3 factorial).

- \( 3! = 3 \times 2 \times 1 = 6 \)

4. **List Possible Arrangements**:
The arrangements for M, O, and P in the positions 2, 3, and 4 can be:
1. MOP
2. MPO
3. OMP
4. OPM
5. PMO
6. POM

5. **Combine with Fixed Positions**:
Each arrangement of the three friends can be combined with N and L fixed in their respective positions:

- N M O L
- N M P L
- N O M L
- N O P L
- N P M L
- N P O L

Thus, the complete arrangements are:
1. NMOPL
2. NMPOL
3. NOMP
4. NOPML
5. NPMOL
6. NPOML

### Final Answer:
Let's compare this with the options provided to find the correct ones:
- NMOPL
- NMPOL
- NOPML
- NPMOL

From the options provided only some of them are present in these arrangements:

1. NMOPL
2. NMPOL
3. NOMPL
4. NPMOL

So, the valid outcomes where Nicole is visited first and Louise is visited last are:
- NMOPL
- NMPOL
- NOPML
- NPMOL

Thus, the correct answer is:
**NMOPL, NMPOL, NOPML, NPMOL**.