The number of ways three different natural numbers cab be drawn from the set `{1, 2, 3, 4,……….., 10}`, if minimum of the chosen numbers is smaller than 4, is

Answer

To solve the problem of finding the number of ways to draw three different natural numbers from the set {1, 2, 3, 4, ..., 10} such that the minimum of the chosen numbers is smaller than 4, we can break it down into three cases based on the minimum number chosen.

### Step-by-step Solution:

1. **Identify the numbers less than 4:**
The numbers in the set that are less than 4 are 1, 2, and 3. Therefore, we will consider three cases based on these minimum values.

2. **Case 1: Minimum number is 1**
- If the minimum number is 1, the other two numbers can be chosen from the set {2, 3, 4, 5, 6, 7, 8, 9, 10}.
- There are 9 numbers in this set (2 to 10).
- We need to choose 2 numbers from these 9.
- The number of ways to choose 2 numbers from 9 is given by the combination formula:
\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]

3. **Case 2: Minimum number is 2**
- If the minimum number is 2, the other two numbers can be chosen from the set {3, 4, 5, 6, 7, 8, 9, 10}.
- There are 8 numbers in this set (3 to 10).
- We need to choose 2 numbers from these 8.
- The number of ways to choose 2 numbers from 8 is:
\[ \binom{8}{2} = \frac{8 \times 7}{2 \times 1} = 28 \]

4. **Case 3: Minimum number is 3**
- If the minimum number is 3, the other two numbers can be chosen from the set {4, 5, 6, 7, 8, 9, 10}.
- There are 7 numbers in this set (4 to 10).
- We need to choose 2 numbers from these 7.
- The number of ways to choose 2 numbers from 7 is:
\[ \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \]

5. **Total number of ways:**
- Now, we add the number of ways from all three cases:
\[ \text{Total} = 36 + 28 + 21 = 85 \]

Thus, the total number of ways to choose three different natural numbers from the set such that the minimum is smaller than 4 is **85**.