There are 8 pencils in a glass, 5 of which are sharpened. Three pencils are randomly taken from the glass simultaneously. We need to find the probability distribution of the number of sharpened pencils among the 3 taken pencils.

1. Problem Description

There are 8 pencils in a glass, 5 of which are sharpened. Three pencils are randomly taken from the glass simultaneously. We need to find the probability distribution of the number of sharpened pencils among the 3 taken pencils.

2. Solution Steps

Let X be the number of sharpened pencils among the 3 pencils taken. Then X can take values 0, 1, 2, or

3. We want to find the probability $P(X = k)$ for $k = 0, 1, 2, 3$.

The total number of ways to choose 3 pencils from 8 is given by the combination formula:

$$$$

C(n, k) = \frac{n!}{k!(n-k)!}

$$$$

The total number of ways to choose 3 pencils from 8 is $C(8, 3) = \frac{8!}{3!5!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$.

Now we calculate the probabilities for each value of X:

$P(X = 0)$: This means we choose 0 sharpened pencils from the 5 sharpened pencils, and 3 unsharpened pencils from the $8 - 5 = 3$ unsharpened pencils.

The number of ways to do this is $C(5, 0) \times C(3, 3) = 1 \times 1 = 1$.

$P(X = 0) = \frac{C(5, 0) \times C(3, 3)}{C(8, 3)} = \frac{1}{56}$.

$P(X = 1)$: This means we choose 1 sharpened pencil from the 5 sharpened pencils, and 2 unsharpened pencils from the 3 unsharpened pencils.

The number of ways to do this is $C(5, 1) \times C(3, 2) = 5 \times 3 = 15$.

$P(X = 1) = \frac{C(5, 1) \times C(3, 2)}{C(8, 3)} = \frac{15}{56}$.

$P(X = 2)$: This means we choose 2 sharpened pencils from the 5 sharpened pencils, and 1 unsharpened pencil from the 3 unsharpened pencils.

The number of ways to do this is $C(5, 2) \times C(3, 1) = \frac{5 \times 4}{2 \times 1} \times 3 = 10 \times 3 = 30$.

$P(X = 2) = \frac{C(5, 2) \times C(3, 1)}{C(8, 3)} = \frac{30}{56} = \frac{15}{28}$.

$P(X = 3)$: This means we choose 3 sharpened pencils from the 5 sharpened pencils, and 0 unsharpened pencils from the 3 unsharpened pencils.

The number of ways to do this is $C(5, 3) \times C(3, 0) = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} \times 1 = 10 \times 1 = 10$.

$P(X = 3) = \frac{C(5, 3) \times C(3, 0)}{C(8, 3)} = \frac{10}{56} = \frac{5}{28}$.

The probability distribution is:

$P(X = 0) = \frac{1}{56}$

$P(X = 1) = \frac{15}{56}$

$P(X = 2) = \frac{30}{56} = \frac{15}{28}$

$P(X = 3) = \frac{10}{56} = \frac{5}{28}$

3. Final Answer

The probability distribution is:

$P(X=0) = 1/56$

$P(X=1) = 15/56$

$P(X=2) = 15/28$

$P(X=3) = 5/28$