Five numbers are selected from 1, 2, 3, 4, 5, 6, 7, 8 and 9. The probability that their product is divisible by 5 or 7 is

Options

A. `(1)/(4)`
B. `(3)/(4)`
C. `(5)/(6)`
D. `(1)/(6)`

Answer

To solve the problem of finding the probability that the product of five numbers selected from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} is divisible by 5 or 7, we can follow these steps:

### Step 1: Calculate Total Ways to Select 5 Numbers
The total number of ways to select 5 numbers from 9 is given by the combination formula:

\[ \text{Total ways} = \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

### Step 2: Calculate Ways to Select Numbers Divisible by 5
To ensure that the product is divisible by 5, we must include the number 5 in our selection. After including 5, we need to select 4 more numbers from the remaining 8 numbers:

\[ \text{Ways to select 4 from 8} = \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

### Step 3: Calculate Ways to Select Numbers Divisible by 7
Similarly, to ensure the product is divisible by 7, we must include the number 7 in our selection. After including 7, we again need to select 4 more numbers from the remaining 8 numbers:

\[ \text{Ways to select 4 from 8} = \binom{8}{4} = 70 \]

### Step 4: Calculate Ways to Select Numbers Divisible by Both 5 and 7
If we want the product to be divisible by both 5 and 7, we must include both 5 and 7 in our selection. We then need to select 3 more numbers from the remaining 7 numbers:

\[ \text{Ways to select 3 from 7} = \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 5: Apply the Inclusion-Exclusion Principle
Now, we can find the total number of ways to select numbers such that the product is divisible by either 5 or 7:

\[ \text{Total ways divisible by 5 or 7} = (\text{Ways divisible by 5}) + (\text{Ways divisible by 7}) - (\text{Ways divisible by both}) \]

Substituting the values we calculated:

\[ \text{Total ways} = 70 + 70 - 35 = 105 \]

### Step 6: Calculate the Probability
Finally, the probability that the product of the selected numbers is divisible by 5 or 7 is given by the ratio of the favorable outcomes to the total outcomes:

\[ \text{Probability} = \frac{\text{Total ways divisible by 5 or 7}}{\text{Total ways to select 5 numbers}} = \frac{105}{126} \]

### Step 7: Simplify the Probability
We can simplify this fraction:

\[ \frac{105}{126} = \frac{5}{6} \]

Thus, the probability that the product of the selected five numbers is divisible by 5 or 7 is:

\[ \text{Probability} = \frac{5}{6} \]