Set A and Set B each consist of 5 distinct numbers. The 2 sets contain identical numbers with the exception of the number with the least value in each set. The number with the least value in Set B is greater than the number with the least value in Set A. The value of which of the following measures must be greater for Set B than for Set `Delta`?

Options

A. Mean only
B. Median only
C. Mode only
D. Mean and median only

Answer

To solve the problem, we need to analyze the information given about Set A and Set B and determine which measure (mean, median, or mode) must be greater for Set B than for Set A.

### Step-by-Step Solution:

1. **Define the Sets**:
- Let Set A = {A, P, Q, R, S} where A is the smallest number.
- Let Set B = {B, P, Q, R, S} where B is the smallest number in Set B and is greater than A.

2. **Understanding the Sets**:
- Both sets contain 5 distinct numbers.
- The only difference between the two sets is the smallest number: A in Set A and B in Set B, with B > A.

3. **Calculate the Mean**:
- The mean of Set A is given by:
\[ \text{Mean of Set A} = \frac{A + P + Q + R + S}{5} \]
- The mean of Set B is given by:
\[ \text{Mean of Set B} = \frac{B + P + Q + R + S}{5} \]

4. **Comparing the Means**:
- Since B > A, we can say:
\[ \text{Mean of Set B} = \frac{B + P + Q + R + S}{5} > \frac{A + P + Q + R + S}{5} = \text{Mean of Set A} \]
- Therefore, the mean of Set B is greater than the mean of Set A.

5. **Calculate the Median**:
- The median of both sets is the middle value when the numbers are arranged in ascending order.
- In both sets, the middle value is Q. Thus:
\[ \text{Median of Set A} = Q \]
\[ \text{Median of Set B} = Q \]
- Therefore, the median is the same for both sets.

6. **Calculate the Mode**:
- The mode is the most frequently occurring number in a set.
- Since both sets consist of distinct numbers, there is no repeating number in either set.
- Thus, both sets have no mode, or we can say that every number occurs once, making the mode undefined or not applicable.

### Conclusion:
The only measure that must be greater for Set B than for Set A is the **mean**. The median and mode are the same for both sets.

### Final Answer:
The measure that must be greater for Set B than for Set A is the **mean**.

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