A number is selected from first 100 natural numbers. The probability that number is perfect square or a perfect cube is

Options

A. `7/50`
B. `3/25`
C. `2/25`
D. `1/25`

Answer

To solve the problem of finding the probability that a number selected from the first 100 natural numbers is either a perfect square or a perfect cube, we can follow these steps:

### Step 1: Identify Perfect Squares
First, we need to find all the perfect squares from the first 100 natural numbers. A perfect square is a number that can be expressed as the square of an integer.

The perfect squares from 1 to 100 are:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)

Thus, the perfect squares are: **1, 4, 9, 16, 25, 36, 49, 64, 81**.

**Count of perfect squares = 9.**

### Step 2: Identify Perfect Cubes
Next, we find all the perfect cubes from the first 100 natural numbers. A perfect cube is a number that can be expressed as the cube of an integer.

The perfect cubes from 1 to 100 are:
- \(1^3 = 1\)
- \(2^3 = 8\)
- \(3^3 = 27\)
- \(4^3 = 64\)

Thus, the perfect cubes are: **1, 8, 27, 64**.

**Count of perfect cubes = 4.**

### Step 3: Identify Overlap (Common Elements)
Now we need to check if there are any numbers that are both perfect squares and perfect cubes. These numbers are perfect sixth powers (since \(n^6 = (n^2)^3 = (n^3)^2\)).

The perfect sixth powers from 1 to 100 are:
- \(1^6 = 1\)
- \(2^6 = 64\)

Thus, the common numbers are: **1, 64**.

**Count of common elements = 2.**

### Step 4: Calculate Total Favorable Outcomes
Using the principle of inclusion-exclusion, we can calculate the total count of favorable outcomes:
- Count of perfect squares = 9
- Count of perfect cubes = 4
- Count of common elements = 2

Total favorable outcomes = (Count of perfect squares) + (Count of perfect cubes) - (Count of common elements)
\[ = 9 + 4 - 2 = 11 \]

### Step 5: Calculate Total Outcomes
The total number of outcomes when selecting from the first 100 natural numbers is:
\[ \text{Total outcomes} = 100 \]

### Step 6: Calculate Probability
Now, we can calculate the probability using the formula:
\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{11}{100} \]

### Final Answer
The probability that a number selected from the first 100 natural numbers is either a perfect square or a perfect cube is:
\[ \frac{11}{100} \text{ or } 0.11 \]

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