Let A and B be two sets containing four and two elements respectively. Then the number of subsets of

Q: Why is the total number of subsets $2^8$?

Great question! For any set with $N$ elements, each element has exactly two possibilities: it is either in a subset or it is not. Since there are $N$ elements, and each has $2$ choices, the total number of combinations is $2 \times 2 \times \dots \times 2$ ($N$ times), which is $2^N$. Here, $N=8$, so $2^8 = 256$.

Q: Why use the complement method?

The complement method is a powerful tool in combinatorics. The question asks for subsets with at least $3$ elements, which means $3, 4, 5, 6, 7,$ or $8$ elements. Calculating these six cases individually is tedious and prone to error. It is much faster to calculate the total ($256$) and subtract the cases we don't want ($0, 1,$ and $2$ elements).

Q: What if the question asked for 'at most 3'?

If the question asked for 'at most 3', you would calculate the number of subsets with $0, 1, 2,$ and $3$ elements directly and sum them up. The complement method is most effective when the number of cases you want to exclude is smaller than the number of cases you want to include.

Question

Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set $A 	imes B$, each having at least three elements is :

Options:

(A) 275

(B) 510

(C) 219

(D) 256

Sigma Percentile · Accepted Answer

### Analyzing the Geometry of Sets

Imagine you are standing before a coordinate plane. On the horizontal axis, you have four distinct points representing set $A$. On the vertical axis, you have two points representing set $B$.

When we form the Cartesian product $A 	imes B$, we are essentially creating a grid of all possible pairs $(a, b)$. Each point in this grid is a unique ordered pair.

Since there are $4$ choices for $a$ and $2$ choices for $b$, the total number of points in our grid is:
$$n(A 	imes B) = n(A) 	imes n(B) = 4 	imes 2 = 8$$
This is our foundational universe of $8$ elements.

### The Power of the Power Set

Now, we want to talk about subsets. A subset is simply a collection of some (or all, or none) of these $8$ elements.

For each of the $8$ elements, we have a binary choice: either it is included in our subset, or it is not. This gives us $2$ choices for each of the $8$ elements.

Therefore, the total number of possible subsets is $2^8$, which equals $256$. This $256$ includes everything from the empty set to the set containing all $8$ elements.

### The Art of the Complement

The problem asks for the number of subsets with at least $3$ elements. This means we are interested in subsets with $3, 4, 5, 6, 7,$ or $8$ elements.

If we were to calculate these one by one, we would be doing a lot of heavy lifting. Instead, let's embrace the elegance of the complement method.

We know that the total number of subsets is the sum of subsets with $0, 1, 2, 3, 4, 5, 6, 7,$ and $8$ elements. Our target is the sum of subsets with $3$ through $8$ elements. Thus, our target is:
$$	ext{Required} = 	ext{Total} - \left( \binom{8}{0} + \binom{8}{1} + \binom{8}{2} ight)$$

### Executing the Calculation

Let's calculate the unwanted subsets. First, the number of subsets with $0$ elements is:
$$\binom{8}{0} = 1$$

Next, the number of subsets with exactly $1$ element is:
$$\binom{8}{1} = 8$$

Finally, the number of subsets with exactly $2$ elements is:
$$\binom{8}{2} = \frac{8 	imes 7}{2 	imes 1} = 28$$

Summing these up, we get $1 + 8 + 28 = 37$ unwanted subsets.

Now, for the final step: we subtract these $37$ from our total of $256$. The result is:
$$256 - 37 = 219$$

We have successfully navigated the complexity of the problem by focusing on what we don't need, leaving us with exactly what we do. The final answer is **219**.

Animated Solution for Mathematics - Sets and Relations: Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set $A \times B$ , each having at least three elements is :

Select Answer:

Visualized Solution

Visualizing Sets $A$ and $B$

Size of the Cartesian Product $A \times B$

Total Subsets of $A \times B$

The "At Least Three" Condition

Identifying the Complement

Subsets with $0$ and $1$ Element

Subsets with $2$ Elements

Total Unwanted Subsets

Finding the Required Subsets

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Geometry of Sets

The Power of the Power Set

The Art of the Complement

Executing the Calculation

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