A subset of 5 elements is chosen from the set of first 15 natural numbers. The probability that at least two of the five numbers are consecutive is `lambda`, then the value of `(22)/(lambda)` is equal to

Answer

To solve the problem, we need to find the probability that at least two of the five chosen numbers from the first 15 natural numbers are consecutive. We will denote this probability as \( \lambda \) and then calculate \( \frac{22}{\lambda} \).

### Step-by-step Solution:

1. **Total Ways to Choose 5 Elements**:
The total number of ways to choose 5 elements from the first 15 natural numbers is given by the combination formula:
\[ \text{Total ways} = \binom{15}{5} \]

2. **Finding Unfavorable Cases**:
We need to find the number of ways to choose 5 elements such that no two elements are consecutive. To do this, we can transform the problem:
- If we choose 5 numbers \( a_1, a_2, a_3, a_4, a_5 \) such that \( a_i \) are the chosen numbers, we can define new variables:
\[ b_i = a_i - (i - 1) \]
- This transformation ensures that \( b_1, b_2, b_3, b_4, b_5 \) are distinct and chosen from the set \( \{1, 2, \ldots, 11\} \) (since we have to leave gaps to ensure no two are consecutive).
- The number of ways to choose these 5 distinct numbers from 11 is:
\[ \text{Unfavorable ways} = \binom{11}{5} \]

3. **Calculating the Probability**:
The probability that at least two of the chosen numbers are consecutive is given by:
\[ \lambda = 1 - \frac{\text{Unfavorable ways}}{\text{Total ways}} = 1 - \frac{\binom{11}{5}}{\binom{15}{5}} \]

4. **Calculating the Combinations**:
We calculate \( \binom{11}{5} \) and \( \binom{15}{5} \):
\[ \binom{11}{5} = \frac{11!}{5!(11-5)!} = \frac{11 \times 10 \times 9 \times 8 \times 7}{5 \times 4 \times 3 \times 2 \times 1} = 462 \]
\[ \binom{15}{5} = \frac{15!}{5!(15-5)!} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003 \]

5. **Substituting Values**:
Now substituting the values into the probability formula:
\[ \lambda = 1 - \frac{462}{3003} \]
To simplify:
\[ \lambda = 1 - \frac{462}{3003} = \frac{3003 - 462}{3003} = \frac{2541}{3003} \]

6. **Finding \( \frac{22}{\lambda} \)**:
Now we need to find \( \frac{22}{\lambda} \):
\[ \frac{22}{\lambda} = \frac{22 \times 3003}{2541} \]
Simplifying this:
\[ \frac{22 \times 3003}{2541} = \frac{66066}{2541} \approx 26 \]

### Final Answer:
Thus, the value of \( \frac{22}{\lambda} \) is \( \boxed{26} \).