If `^18C_15+2 ^18C_16+^17C_16+1=^nC_3`, then `n=`

Options

A. 18
B. 20
C. 22
D. 24

Answer

To solve the problem, we need to simplify the expression \( ^{18}C_{15} + 2 \cdot ^{18}C_{16} + ^{17}C_{16} + 1 \) and set it equal to \( ^nC_3 \) to find the value of \( n \).

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ ^{18}C_{15} + 2 \cdot ^{18}C_{16} + ^{17}C_{16} + 1 \]
can be rewritten as:
\[ ^{18}C_{15} + ^{18}C_{16} + ^{18}C_{16} + ^{17}C_{16} + 1 \]
Here, \( 1 \) can be expressed as \( ^{17}C_{17} \).

2. **Apply the combination property**:
We use the property \( ^nC_r + ^nC_{r+1} = ^{n+1}C_{r+1} \).
- First, apply it to \( ^{18}C_{15} + ^{18}C_{16} \):
\[ ^{18}C_{15} + ^{18}C_{16} = ^{19}C_{16} \]
- Now, we have:
\[ ^{19}C_{16} + ^{18}C_{16} + ^{17}C_{17} \]

3. **Combine \( ^{19}C_{16} + ^{18}C_{16} \)**:
Again, apply the property:
\[ ^{19}C_{16} + ^{18}C_{16} = ^{19}C_{16} + ^{19}C_{17} = ^{20}C_{17} \]

4. **Combine \( ^{20}C_{17} + ^{17}C_{17} \)**:
Now, apply the property again:
\[ ^{20}C_{17} + ^{17}C_{17} = ^{20}C_{17} + 1 = ^{20}C_{17} + ^{20}C_{18} = ^{21}C_{18} \]

5. **Express \( ^{21}C_{18} \) in terms of \( nC_3 \)**:
We can express \( ^{21}C_{18} \) as:
\[ ^{21}C_{18} = ^{21}C_{3} \]
because \( ^nC_r = ^nC_{n-r} \).

6. **Set the equation**:
Now we have:
\[ ^{n}C_{3} = ^{21}C_{3} \]
This implies that \( n = 21 \).

### Final Answer:
Thus, the value of \( n \) is:
\[ \boxed{21} \]