If 4 distinct numbers are chosen randomly from the first 100 natural numbers, then the probability that all 4 of them are either divisible by 3 or divisible by 5 is

Options

A. `(.^(6)C_(4))/(^(.100)C_(4))`
B. `(.^(33)C_(4))/(.^(100)C_(4))`
C. `(.^(20)C_(4))/(.^(100)C_(4))`
D. `(.^(47)C_(4))/(.^(100)C_(4))`

Answer

To solve the problem, we need to find the probability that all 4 distinct numbers chosen from the first 100 natural numbers are either divisible by 3 or divisible by 5. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the total number of ways to choose 4 distinct numbers from 100.
The total number of ways to choose 4 distinct numbers from 100 can be calculated using the combination formula:
\[ \text{Total ways} = \binom{100}{4} \]

### Step 2: Count the numbers divisible by 3.
To find how many numbers from 1 to 100 are divisible by 3, we can use the formula:
\[ \text{Numbers divisible by 3} = \left\lfloor \frac{100}{3} \right\rfloor = 33 \]

### Step 3: Count the numbers divisible by 5.
Similarly, for numbers divisible by 5:
\[ \text{Numbers divisible by 5} = \left\lfloor \frac{100}{5} \right\rfloor = 20 \]

### Step 4: Count the numbers divisible by both 3 and 5 (i.e., divisible by 15).
To avoid double counting, we need to find the numbers that are divisible by both 3 and 5:
\[ \text{Numbers divisible by 15} = \left\lfloor \frac{100}{15} \right\rfloor = 6 \]

### Step 5: Use the principle of inclusion-exclusion to find the total numbers divisible by either 3 or 5.
Using the principle of inclusion-exclusion:
\[ \text{Numbers divisible by 3 or 5} = (\text{Numbers divisible by 3}) + (\text{Numbers divisible by 5}) - (\text{Numbers divisible by 15}) \]
\[ = 33 + 20 - 6 = 47 \]

### Step 6: Determine the number of favorable outcomes.
Now, we need to choose 4 distinct numbers from the 47 numbers that are either divisible by 3 or 5:
\[ \text{Favorable ways} = \binom{47}{4} \]

### Step 7: Calculate the probability.
The probability that all 4 chosen numbers are either divisible by 3 or 5 is given by:
\[ \text{Probability} = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{\binom{47}{4}}{\binom{100}{4}} \]

### Step 8: Final Calculation.
Now we can calculate the values of \(\binom{47}{4}\) and \(\binom{100}{4}\) to find the exact probability.