If `""^(18)C_(15)+2(""^(18)C_(16))+""^(17)C_(16)+1=""^(n)C_(3)`, then n is equal to :

Options

A. 19
B. 20
C. 18
D. 24

Answer

To solve the equation \( \binom{18}{15} + 2 \binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3} \), we will simplify the left-hand side step by step.

### Step 1: Simplify the Left-Hand Side
We start with the left-hand side:
\[ \binom{18}{15} + 2 \binom{18}{16} + \binom{17}{16} + 1 \]

### Step 2: Use Binomial Coefficient Properties
We can use the property of binomial coefficients that states:
\[ \binom{n}{k} = \binom{n}{n-k} \]
Thus, we can rewrite:
\[ \binom{18}{15} = \binom{18}{3} \quad \text{and} \quad \binom{18}{16} = \binom{18}{2} \quad \text{and} \quad \binom{17}{16} = \binom{17}{1} \]

### Step 3: Substitute the Values
Now substituting these into the equation:
\[ \binom{18}{3} + 2 \binom{18}{2} + \binom{17}{1} + 1 \]

### Step 4: Calculate Each Binomial Coefficient
Now we compute each binomial coefficient:
1. \(\binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 816\)
2. \(\binom{18}{2} = \frac{18 \times 17}{2 \times 1} = 153\)
3. \(\binom{17}{1} = 17\)

### Step 5: Substitute Back into the Equation
Now substituting these values back into the equation:
\[ 816 + 2 \times 153 + 17 + 1 \]
Calculating \(2 \times 153 = 306\):
\[ 816 + 306 + 17 + 1 = 1140 \]

### Step 6: Set the Left-Hand Side Equal to the Right-Hand Side
Now we have:
\[ 1140 = \binom{n}{3} \]

### Step 7: Solve for \(n\)
To find \(n\), we need to solve:
\[ \binom{n}{3} = \frac{n(n-1)(n-2)}{6} = 1140 \]
Multiplying both sides by 6:
\[ n(n-1)(n-2) = 6840 \]

### Step 8: Find \(n\)
We can test values for \(n\):
- For \(n = 20\):
\[ 20 \times 19 \times 18 = 6840 \]
Thus, \(n = 20\) satisfies the equation.

### Final Answer
Therefore, the value of \(n\) is:
\[ \boxed{20} \]