Explanation

If a set contains n elements, then the number of subsets of this set is equal to 2ⁿ − 1.

The only subset which is not proper is the set itself.

So, to get the number of proper subsets, you just need to subtract one from the total number of subsets.

Explanation

A conjunction p ∧ q (p AND q) is true only when both p and q are true.

Explanation

The symbol ∼p means "not p".

If ∼p is true, then p must be false.

Explanation

• The sum of deviations from the regression line is zero.
• This is a fundamental property of linear regression, known as the "zero-sum property" or "error-sum property".

Explanation

• When 3 coins are tossed, there are 2^3 = 8 possible outcomes.
• Only one of these outcomes is TTT (3 tails), so the probability is 1/8.

Explanation

To solve this, we can follow these steps:

• Given:

  • Gross Profit = Rs. 5,000

  • Net Profit = 25% of Gross Profit

Step 1: Calculate the Net Profit.

Since the $text{Net Profit} = 25$
$text{Net Profit} = 0.25 times 5000 = Rs. 1,250$

Step 2: Calculate the Expenses.

The relationship between Gross Profit, Net Profit, and Expenses is:

$$\text{Gross Profit}=\text{Net Profit}+\text{Expenses}$$

Rearranging this formula to find the expenses:

$$\text{Expenses}=\text{Gross Profit}-\text{Net Profit}$$

Now, substitute the values:

$$\text{Expenses}=5000-1250=Rs.3,750$$

Explanation

Number of absences is a countable quantity (0, 1, 2, ...), so it's discrete data.

Discrete data deals with whole numbers, not fractions or decimals.

Explanation

In the presence of multicollinearity, the OLS estimator will still be consistent unbiased and efficient.

Explanation

There are 12 face cards in a standard deck of 52 cards (3 face cards per suit x 4 suits).

The probability of drawing 2 face cards is: C(12,2) x C(40,3) / C(52,5) = 0.0533

Explanation

The variance of a sample is calculated as the square of the standard deviation. Since the standard deviation is 64, the variance is:

Variance = σ^2 = 64^2 = 4096

So, the correct answer is 4096.