If `((n),(r+1))=56, ((n),(r ))=28 and ((n),(r-1))=8`, then `n+r` is equal to

Options

A. 8
B. 10
C. 12
D. 9

Answer

To solve the problem, we are given three equations involving binomial coefficients:

1. \(\binom{n}{r+1} = 56\)
2. \(\binom{n}{r} = 28\)
3. \(\binom{n}{r-1} = 8\)

We need to find the value of \(n + r\).

### Step 1: Use the relationship between binomial coefficients
We know the relationship between consecutive binomial coefficients:
\[ \binom{n}{r+1} = \frac{n - r}{r + 1} \cdot \binom{n}{r} \]
Using this, we can express \(\binom{n}{r+1}\) in terms of \(\binom{n}{r}\):
\[ 56 = \frac{n - r}{r + 1} \cdot 28 \]
Dividing both sides by 28:
\[ \frac{n - r}{r + 1} = 2 \quad \text{(Equation 1)} \]

### Step 2: Use another relationship for \(\binom{n}{r-1}\)
Similarly, we can express \(\binom{n}{r}\) in terms of \(\binom{n}{r-1}\):
\[ \binom{n}{r} = \frac{r}{n - r + 1} \cdot \binom{n}{r-1} \]
Using this, we can express \(\binom{n}{r}\):
\[ 28 = \frac{r}{n - r + 1} \cdot 8 \]
Dividing both sides by 8:
\[ \frac{r}{n - r + 1} = 3.5 \quad \text{(Equation 2)} \]

### Step 3: Solve Equation 1 for \(n\)
From Equation 1:
\[ n - r = 2(r + 1) \]
Expanding this gives:
\[ n - r = 2r + 2 \]
Rearranging gives:
\[ n = 3r + 2 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 3 into Equation 2
Substituting \(n\) from Equation 3 into Equation 2:
\[ \frac{r}{(3r + 2) - r + 1} = 3.5 \]
This simplifies to:
\[ \frac{r}{2r + 3} = 3.5 \]
Cross-multiplying gives:
\[ r = 3.5(2r + 3) \]
Expanding this:
\[ r = 7r + 10.5 \]
Rearranging gives:
\[ -6r = 10.5 \quad \Rightarrow \quad r = -\frac{10.5}{6} = -1.75 \quad \text{(not valid)} \]

### Step 5: Re-evaluate the equations
Let's go back to Equation 1 and Equation 2:
From Equation 1:
\[ n - r = 2r + 2 \quad \Rightarrow \quad n = 3r + 2 \]

From Equation 2:
\[ \frac{r}{n - r + 1} = 3.5 \]
Substituting \(n = 3r + 2\):
\[ \frac{r}{(3r + 2) - r + 1} = 3.5 \quad \Rightarrow \quad \frac{r}{2r + 3} = 3.5 \]
Cross-multiplying gives:
\[ r = 3.5(2r + 3) \quad \Rightarrow \quad r = 7r + 10.5 \]
Rearranging gives:
\[ -6r = 10.5 \quad \Rightarrow \quad r = 2 \]

### Step 6: Find \(n\)
Substituting \(r = 2\) back into Equation 3:
\[ n = 3(2) + 2 = 6 + 2 = 8 \]

### Step 7: Calculate \(n + r\)
Finally, we find:
\[ n + r = 8 + 2 = 10 \]

### Final Answer
Thus, \(n + r = 10\).