Match the entries of List-I with the correct entries of List-II. Which of the following option is correct?

Steps to Solve

1. Calculate the number of subsets of E containing at least one even number (P)

To find the total number of subsets of a set with 7 elements, use the formula $2^n$, where $n$ is the number of elements: $$ 2^7 = 128 $$ Now, calculate the number of subsets with only odd numbers. The odd elements in set $E$ are ${1, 3, 5, 7}$, which has 4 elements, so the number of subsets is: $$ 2^4 = 16 $$ Thus, the number of subsets containing at least one even number is: $$ 128 - 16 = 112 $$

2. Determine the number of positive integral solutions for $y_1 \cdot y_2 \cdot y_3 = 105$ (Q)

First, factorize 105 into primes: $$ 105 = 3 \times 5 \times 7 $$ Using the "stars and bars" method, the number of factors (exponents) is 3. For each prime factor $p^k$, the number of ways to choose $k$ from $n$ where $k \geq 0$ is given by $(k + n - 1)! / (k! \cdot (n - 1)!)$ which leads to: $$ (3 + 3 - 1)! / (3! \cdot (3 - 1)!) = \frac{5!}{3! \cdot 2!} = 10 $$ Thus, the required number of solutions is $10$.

3. Count the six-digit numbers where each digit exceeds the previous (R)

The digits must be chosen from ${1, 2, 3, 4, 5, 6, 7, 8, 9}$, choosing 6 different digits. The number of ways to select 6 digits from 9 is given by: $$ \binom{9}{6} = 84 $$

4. Find the number of functions from ${1, 2, 3, 4}$ to ${1, 2, 3, 4, 5}$ (S)

Each of the 4 inputs can map to any of the 5 outputs, so the total number of functions is: $$ 5^4 = 625 $$