Mean Value of Difference in Chosen Books Count

To find the mean value $\alpha$ of the random variable $X$, we first define the total number of ways to choose 6 books out of 11 (6 Mathematics books and 5 Physics books).

The total number of ways to select 6 books from 11 is given by: $N_{\text{total}} = \binom{11}{6} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462$

Let $m$ denote the number of Mathematics books chosen, and $p$ denote the number of Physics books chosen. Since a total of 6 books are selected, we have $m + p = 6$.

The random variable $X$ represents the absolute difference between $m$ and $p$: $X = |m - p| = |m - (6 - m)| = |2m - 6|$

Since there are 6 Mathematics books and 5 Physics books available, the possible values for $m$ are $m \in \{1, 2, 3, 4, 5, 6\}$.

Now, we calculate the number of ways, the value of $X$, and the corresponding probabilities for each possible value of $m$:

1. For $m = 1$ ($p = 5$):

   • $X = |1 - 5| = 4$
   • Number of ways = $\binom{6}{1} \binom{5}{5} = 6 \times 1 = 6$

2. For $m = 2$ ($p = 4$):

   • $X = |2 - 4| = 2$
   • Number of ways = $\binom{6}{2} \binom{5}{4} = 15 \times 5 = 75$

3. For $m = 3$ ($p = 3$):

   • $X = |3 - 3| = 0$
   • Number of ways = $\binom{6}{3} \binom{5}{3} = 20 \times 10 = 200$

4. For $m = 4$ ($p = 2$):

   • $X = |4 - 2| = 2$
   • Number of ways = $\binom{6}{4} \binom{5}{2} = 15 \times 10 = 150$

5. For $m = 5$ ($p = 1$):

   • $X = |5 - 1| = 4$
   • Number of ways = $\binom{6}{5} \binom{5}{1} = 6 \times 5 = 30$

6. For $m = 6$ ($p = 0$):

   • $X = |6 - 0| = 6$
   • Number of ways = $\binom{6}{6} \binom{5}{0} = 1 \times 1 = 1$

The mean $\alpha = E[X]$ is given by: $\alpha = \sum X \cdot P(X) = \frac{1}{N_{\text{total}}} \sum (X \times \text{Number of ways})$

Substituting the values into the formula: $\alpha = \frac{1}{462} \left[ (4 \times 6) + (2 \times 75) + (0 \times 200) + (2 \times 150) + (4 \times 30) + (6 \times 1) \right]$

$\alpha = \frac{1}{462} \left[ 24 + 150 + 0 + 300 + 120 + 6 \right]$

$\alpha = \frac{600}{462}$

Simplifying the fraction by dividing both numerator and denominator by 6: $\alpha = \frac{100}{77}$

We are required to find the value of $77\alpha$: $77\alpha = 77 \times \frac{100}{77} = 100$