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 calculate the probability that all four distinct numbers chosen from the first 100 natural numbers are either divisible by 3 or divisible by 5. Here’s a step-by-step solution:

### Step 1: Calculate the Total Number of Ways to Choose 4 Numbers
The total number of ways to choose 4 distinct numbers from the first 100 natural numbers is given by the combination formula:

\[ \text{Total cases} = \binom{100}{4} \]

### Step 2: Find the Count of Numbers Divisible by 3
To find how many numbers from 1 to 100 are divisible by 3, we can use the formula:

\[ \text{Count of numbers divisible by 3} = \left\lfloor \frac{100}{3} \right\rfloor = 33 \]

### Step 3: Find the Count of Numbers Divisible by 5
Similarly, for numbers divisible by 5:

\[ \text{Count of numbers divisible by 5} = \left\lfloor \frac{100}{5} \right\rfloor = 20 \]

### Step 4: Find the Count of Numbers Divisible by Both 3 and 5
Numbers that are divisible by both 3 and 5 are those divisible by their least common multiple (LCM), which is 15:

\[ \text{Count of numbers divisible by 15} = \left\lfloor \frac{100}{15} \right\rfloor = 6 \]

### Step 5: Apply the Inclusion-Exclusion Principle
To find the total count of numbers that are divisible by either 3 or 5, we use the inclusion-exclusion principle:

\[ \text{Count of numbers divisible by 3 or 5} = \text{Count divisible by 3} + \text{Count divisible by 5} - \text{Count divisible by both} \]

Substituting the values we found:

\[ \text{Count} = 33 + 20 - 6 = 47 \]

### Step 6: Calculate the Number of Favorable Cases
The number of favorable cases (choosing 4 numbers from those that are either divisible by 3 or 5) is:

\[ \text{Favorable cases} = \binom{47}{4} \]

### Step 7: Calculate the Probability
The probability that all four chosen numbers are either divisible by 3 or 5 is given by the ratio of favorable cases to total cases:

\[ \text{Probability} = \frac{\binom{47}{4}}{\binom{100}{4}} \]

### Final Answer
Thus, the probability that all four numbers chosen are either divisible by 3 or divisible by 5 is:

\[ \frac{\binom{47}{4}}{\binom{100}{4}} \]