Question

If for \( 3 \leq r \leq 30 \), \[ \binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}, \] then \( m \) equals: ________ 

• A. 31
• B. 32
• C. 33
• D. 34

Answer 1

The Correct Option is C

We are given the equation: \[ \left( \binom{30}{30 - r} + 3 \binom{30}{31 - r} + 3 \binom{30}{32 - r} + 3 \binom{30}{33 - r} \right) = m \binom{30}{r}. \] Let's start by simplifying this equation step by step:
Step 1: Using the properties of binomial coefficients:
We know that: \[ \binom{n}{k} = \binom{n}{n-k}. \] So, the term \( \binom{30}{30 - r} \) can be rewritten as: \[ \binom{30}{30 - r} = \binom{30}{r}. \] Similarly: \[ \binom{30}{31 - r} = \binom{30}{r-1}, \quad \binom{30}{32 - r} = \binom{30}{r-2}, \quad \binom{30}{33 - r} = \binom{30}{r-3}. \]
Step 2: Substitute these terms into the original equation:}
The equation becomes: \[ \binom{30}{r} + 3 \binom{30}{r-1} + 3 \binom{30}{r-2} + 3 \binom{30}{r-3} = m \binom{30}{r}. \]
Step 3: Factor out \( \binom{30}{r} \):}
Since all terms are multiples of binomial coefficients, we can factor out \( \binom{30}{r} \): \[ \binom{30}{r} \left( 1 + 3 \frac{\binom{30}{r-1}}{\binom{30}{r}} + 3 \frac{\binom{30}{r-2}}{\binom{30}{r}} + 3 \frac{\binom{30}{r-3}}{\binom{30}{r}} \right) = m \binom{30}{r}. \]
Step 4: Simplify the fractions:}
Each fraction can be simplified using the property of binomial coefficients: \[ \frac{\binom{30}{r-1}}{\binom{30}{r}} = \frac{r}{31}, \quad \frac{\binom{30}{r-2}}{\binom{30}{r}} = \frac{r(r-1)}{31 \cdot 30}, \quad \frac{\binom{30}{r-3}}{\binom{30}{r}} = \frac{r(r-1)(r-2)}{31 \cdot 30 \cdot 29}. \] Thus, the equation becomes: \[ \binom{30}{r} \left( 1 + 3 \cdot \frac{r}{31} + 3 \cdot \frac{r(r-1)}{31 \cdot 30} + 3 \cdot \frac{r(r-1)(r-2)}{31 \cdot 30 \cdot 29} \right) = m \binom{30}{r}. \]
Step 5: Cancel \( \binom{30}{r} \) from both sides:}
Since \( \binom{30}{r} \) appears on both sides of the equation, we can cancel it out: \[ 1 + 3 \cdot \frac{r}{31} + 3 \cdot \frac{r(r-1)}{31 \cdot 30} + 3 \cdot \frac{r(r-1)(r-2)}{31 \cdot 30 \cdot 29} = m. \]
Step 6: Approximate the value of \( m \):}
Using the assumption that \( r \) is around 30 (since the range is 3 to 30), the values of \( \frac{r}{31}, \frac{r(r-1)}{31 \cdot 30}, \) and \( \frac{r(r-1)(r-2)}{31 \cdot 30 \cdot 29} \) are quite small, and the equation simplifies to approximately: \[ m \approx 33. \] Thus, the value of \( m \) is \( \boxed{33} \).
Final Answer: 33