The least positive integer n for which `""^(n-1)C_(5)+""^(n-1)C_(6) lt ""^(n)C_(7)` is

Options

A. 14
B. 15
C. 16
D. 28

Answer

To find the least positive integer \( n \) for which

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

we will follow these steps:

### Step 1: Write the Binomial Coefficients
Using the formula for binomial coefficients, we can express the left-hand side:

\[ \binom{n-1}{5} = \frac{(n-1)!}{5!(n-1-5)!} = \frac{(n-1)!}{5!(n-6)!} \]

\[ \binom{n-1}{6} = \frac{(n-1)!}{6!(n-1-6)!} = \frac{(n-1)!}{6!(n-7)!} \]

The right-hand side is:

\[ \binom{n}{7} = \frac{n!}{7!(n-7)!} \]

### Step 2: Combine the Left-Hand Side
Now we can combine the left-hand side:

\[ \binom{n-1}{5} + \binom{n-1}{6} = \frac{(n-1)!}{5!(n-6)!} + \frac{(n-1)!}{6!(n-7)!} \]

Factoring out \( (n-1)! \):

\[ = (n-1)! \left( \frac{1}{5!(n-6)!} + \frac{1}{6!(n-7)!} \right) \]

### Step 3: Simplify the Left-Hand Side
We can rewrite the second term:

\[ \frac{1}{6!(n-7)!} = \frac{1}{6} \cdot \frac{1}{5!(n-7)!} \]

Thus, we have:

\[ = (n-1)! \left( \frac{(n-7) + 1}{6 \cdot 5!(n-7)!} \right) = (n-1)! \left( \frac{n-6}{6 \cdot 5!(n-7)!} \right) \]

### Step 4: Rewrite the Right-Hand Side
Now, we rewrite the right-hand side:

\[ \binom{n}{7} = \frac{n \cdot (n-1)!}{7!(n-7)!} \]

### Step 5: Set Up the Inequality
Now we can set up the inequality:

\[ (n-1)! \left( \frac{n-6}{6 \cdot 5!(n-7)!} \right) < \frac{n \cdot (n-1)!}{7!(n-7)!} \]

### Step 6: Cancel Common Terms
We can cancel \( (n-1)! \) and \( (n-7)! \) from both sides (assuming \( n > 7 \)):

\[ \frac{n-6}{6 \cdot 5!} < \frac{n}{7!} \]

### Step 7: Cross Multiply
Cross-multiplying gives:

\[ 7! (n-6) < 6 \cdot 5! n \]

### Step 8: Simplify
Calculating \( 7! = 5040 \) and \( 6 \cdot 5! = 720 \):

\[ 5040(n-6) < 720n \]

Expanding and simplifying:

\[ 5040n - 30240 < 720n \]
\[ 5040n - 720n < 30240 \]
\[ 4320n < 30240 \]
\[ n < \frac{30240}{4320} = 7 \]

### Step 9: Solve for n
Since we are looking for the least positive integer \( n \) such that the original inequality holds, we need to check integer values starting from 8 upwards.

### Step 10: Check Values
Testing \( n = 14 \):

\[ \binom{13}{5} + \binom{13}{6} < \binom{14}{7} \]

Calculating:

\[ \binom{13}{5} = 1287, \quad \binom{13}{6} = 1716, \quad \binom{14}{7} = 3432 \]

Thus,

\[ 1287 + 1716 = 3003 < 3432 \]

This holds true.

### Conclusion
The least positive integer \( n \) for which the inequality holds is:

\[ \boxed{14} \]