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 find the probability that the product of five selected numbers 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 the Total Number of Outcomes
The total number of ways to select 5 numbers from the set of 9 numbers is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Total outcomes} = \binom{9}{5} \]

### Step 2: Identify Favorable Outcomes
We need to find the number of favorable outcomes where the product of the selected numbers is divisible by 5 or 7.

#### Case 1: Product is Divisible by 5
For the product to be divisible by 5, at least one of the selected numbers must be 5. If we fix the number 5, we need to choose 4 more numbers from the remaining 8 numbers.

\[ N(A) = \binom{8}{4} \]

#### Case 2: Product is Divisible by 7
For the product to be divisible by 7, at least one of the selected numbers must be 7. If we fix the number 7, we need to choose 4 more numbers from the remaining 8 numbers.

\[ N(B) = \binom{8}{4} \]

#### Case 3: Product is Divisible by Both 5 and 7
If the product is divisible by both 5 and 7, we must include both 5 and 7 in our selection. We then need to choose 3 more numbers from the remaining 7 numbers.

\[ N(A \cap B) = \binom{7}{3} \]

### Step 3: Use the Inclusion-Exclusion Principle
To find the total number of favorable outcomes (where the product is divisible by 5 or 7), we use the inclusion-exclusion principle:

\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]

Substituting the values we calculated:

\[ N(A \cup B) = \binom{8}{4} + \binom{8}{4} - \binom{7}{3} \]

### Step 4: Calculate the Combinations
Now we calculate the combinations:

\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]
\[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

So,

\[ N(A \cup B) = 70 + 70 - 35 = 105 \]

### Step 5: Calculate the Probability
Now, we can find the probability that the product of the selected numbers is divisible by 5 or 7:

\[ P(\text{Divisible by 5 or 7}) = \frac{N(A \cup B)}{\text{Total outcomes}} = \frac{105}{\binom{9}{5}} \]

Calculating \( \binom{9}{5} \):

\[ \binom{9}{5} = \binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

Thus,

\[ P(\text{Divisible by 5 or 7}) = \frac{105}{126} = \frac{5}{6} \]

### Final Answer
The probability that the product of the selected numbers is divisible by 5 or 7 is:

\[ \frac{5}{6} \]