Sets

Step 1:


Step 2: Since and are not fixed, multiple values are possible.

Step 3: Valid arrangements give 26, 27, or 28.

Using the formula:

n(A∪ B∪ C)=n(A)+n(B)+n(C)-n(A∩ B)-n(B∩ C)-n(A∩ C)+n(A∩ B∩ C) =10+15+20-8-6-n(A∩ C)+n(A∩ B∩ C) =31-bigl[n(A∩ C)-n(A∩ B∩ C)bigr]

Since 0≤ n(A∩ B∩ C)≤ n(A∩ C)≤ min(10,20)=10, the quantity n(A∩ C)-n(A∩ B∩ C) can take values 3,4,5. Hence, n(A∪ B∪ C)=31-5,31-4,31-3=26,27,28 So all three values are possible.

Step 1: {Identify the distinct letters}
The word 'MONOTONE' has the following distinct letters: M, O, N, T, E.
Step 2: {Count the distinct letters}
There are 5 distinct letters.
Step 3: {Use the formula for the number of subsets}
The number of subsets of a set with elements is . In this case, .
Step 4: {Calculate the number of subsets}
Number of subsets = .

We are given and the condition that are distinct. The goal is to find which of the following expressions is greater than.
Step 1: Use AM-GM inequality
We know that the Arithmetic Mean is greater than or equal to the Geometric Mean: Applying this to the terms , we get: Thus, the correct answer is , which matches option (D).

To find the intersection of the sets and , we must first determine the intervals represented by each set.

Set A:

We solve the absolute value inequality:

This implies:

Adding 2 to all sides:

So, .

Set B:

We solve the absolute value inequality:

This implies:

Subtracting 1 from all sides:

So, .

Intersection :

To find , take the overlap of the intervals and .

The overlap is from to , where is not included but is included in set .

Hence, .