If ` .^(20)C_(n+1) = ^ (n) C_(16)`, then the value of n is a.7 b. 10 c. 13 d. None of these

Options

A. 7
B. 10
C. 13
D. None of these

Answer

To solve the equation \( \binom{20}{n+1} = \binom{n}{16} \), we will follow these steps:

### Step 1: Understand the conditions for \( n \)
The left-hand side \( \binom{20}{n+1} \) is defined only if \( n+1 \leq 20 \), which implies:
\[ n \leq 19 \]
The right-hand side \( \binom{n}{16} \) is defined only if \( n \geq 16 \). Therefore, we have:
\[ n \geq 16 \]
Combining these two inequalities, we find:
\[ 16 \leq n \leq 19 \]

### Step 2: Check possible values of \( n \)
The possible integer values for \( n \) are 16, 17, 18, and 19. We will check each of these values to see if they satisfy the equation.

#### Check \( n = 16 \):
\[ \binom{20}{16+1} = \binom{20}{17} \quad \text{and} \quad \binom{16}{16} = 1 \]
Calculating:
\[ \binom{20}{17} = \binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140 \]
So, \( 1140 \neq 1 \).

#### Check \( n = 17 \):
\[ \binom{20}{17+1} = \binom{20}{18} \quad \text{and} \quad \binom{17}{16} = 17 \]
Calculating:
\[ \binom{20}{18} = \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190 \]
So, \( 190 \neq 17 \).

#### Check \( n = 18 \):
\[ \binom{20}{18+1} = \binom{20}{19} \quad \text{and} \quad \binom{18}{16} = \binom{18}{2} = \frac{18 \times 17}{2 \times 1} = 153 \]
Calculating:
\[ \binom{20}{19} = \binom{20}{1} = 20 \]
So, \( 20 \neq 153 \).

#### Check \( n = 19 \):
\[ \binom{20}{19+1} = \binom{20}{20} \quad \text{and} \quad \binom{19}{16} = \binom{19}{3} = \frac{19 \times 18 \times 17}{3 \times 2 \times 1} = 969 \]
Calculating:
\[ \binom{20}{20} = 1 \]
So, \( 1 \neq 969 \).

### Conclusion:
None of the values \( n = 16, 17, 18, 19 \) satisfy the equation \( \binom{20}{n+1} = \binom{n}{16} \). Therefore, the answer is:
\[ \text{Option D: None of these} \]

---