The least value of natural number n such that `((n-1),(5))+((n-1),(6))lt((n),(7))," where"((n),(r))=(n!)/((n-r)!r!),is `

Options

A. 11
B. 10
C. 12
D. 13

Answer

To solve the problem, we need to find the least value of the natural number \( n \) such that:

\[ \binom{n-1}{5} + \binom{n-1}{6} < \binom{n}{7} \]

where \( \binom{n}{r} = \frac{n!}{(n-r)!r!} \).

### Step-by-Step Solution:

1. **Express the Binomial Coefficients**:
Using the formula for binomial coefficients, we can express the left-hand side and the right-hand side as follows:
\[ \binom{n-1}{5} = \frac{(n-1)!}{5!(n-6)!} \]
\[ \binom{n-1}{6} = \frac{(n-1)!}{6!(n-7)!} \]
\[ \binom{n}{7} = \frac{n!}{7!(n-7)!} \]

2. **Combine the Left-Hand Side**:
The left-hand side becomes:
\[ \frac{(n-1)!}{5!(n-6)!} + \frac{(n-1)!}{6!(n-7)!} \]
To combine these, we can factor out \( (n-1)! \):
\[ (n-1)! \left( \frac{1}{5!(n-6)!} + \frac{1}{6!(n-7)!} \right) \]

The second term can be rewritten as:
\[ \frac{1}{6!(n-7)!} = \frac{1}{6} \cdot \frac{1}{5!(n-7)!} \]
Thus, we have:
\[ (n-1)! \left( \frac{1}{5!(n-6)!} + \frac{1}{6} \cdot \frac{1}{5!(n-7)!} \right) \]

3. **Simplify the Left-Hand Side**:
We can simplify further:
\[ = (n-1)! \left( \frac{(n-7) + \frac{1}{6}}{6 \cdot 5!(n-7)!} \right) \]
This results in:
\[ = \frac{(n-1)!}{5!(n-7)!} \left( \frac{n-6}{6} \right) \]

4. **Right-Hand Side**:
The right-hand side can be expressed as:
\[ \binom{n}{7} = \frac{n \cdot (n-1)!}{7!(n-7)!} \]

5. **Set Up the Inequality**:
Now we can set up the inequality:
\[ \frac{(n-1)!}{5!(n-7)!} \cdot \frac{n-6}{6} < \frac{n \cdot (n-1)!}{7!(n-7)!} \]

6. **Cancel Common Terms**:
Cancel \( (n-1)! \) and \( (n-7)! \) from both sides:
\[ \frac{n-6}{6 \cdot 5!} < \frac{n}{7!} \]

7. **Cross Multiply**:
Cross-multiply to eliminate the fractions:
\[ 7! (n-6) < 6 \cdot 5! n \]
Simplifying gives:
\[ 5040(n-6) < 720n \]

8. **Rearrange the Inequality**:
Rearranging leads to:
\[ 5040n - 30240 < 720n \]
\[ 5040n - 720n < 30240 \]
\[ 4320n < 30240 \]
\[ n < \frac{30240}{4320} = 7 \]

9. **Find the Least Natural Number**:
The inequality \( n > 13 \) gives us the least natural number \( n = 14 \).

### Final Answer:
The least value of the natural number \( n \) that satisfies the inequality is:

\[ \boxed{14} \]