If the coefficient of 4th term in the expansion of `(a+b)^n` is 56, then n is

Explanation

To solve the problem, we need to find the value of \( n \) such that the coefficient of the 4th term in the expansion of \( (a + b)^n \) is 56.

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
where \( T_{r+1} \) is the \( (r+1)^{th} \) term, \( \binom{n}{r} \) is the binomial coefficient, and \( r \) starts from 0.

2. **Finding the 4th Term**:
The 4th term corresponds to \( r = 3 \) (since \( r \) starts from 0). Therefore, the 4th term is:
\[ T_4 = \binom{n}{3} a^{n-3} b^3 \]
The coefficient of the 4th term is \( \binom{n}{3} \).

3. **Setting Up the Equation**:
According to the problem, the coefficient of the 4th term is given as 56. Thus, we have:
\[ \binom{n}{3} = 56 \]

4. **Expanding the Binomial Coefficient**:
The binomial coefficient \( \binom{n}{3} \) can be expressed as:
\[ \binom{n}{3} = \frac{n(n-1)(n-2)}{3!} = \frac{n(n-1)(n-2)}{6} \]
Therefore, we can set up the equation:
\[ \frac{n(n-1)(n-2)}{6} = 56 \]

5. **Multiplying Both Sides by 6**:
To eliminate the fraction, multiply both sides by 6:
\[ n(n-1)(n-2) = 336 \]

6. **Finding the Value of \( n \)**:
Now we need to find \( n \) such that \( n(n-1)(n-2) = 336 \). We can test integer values for \( n \):

- For \( n = 7 \):
\[ 7 \cdot 6 \cdot 5 = 210 \quad (\text{too low}) \]
- For \( n = 8 \):
\[ 8 \cdot 7 \cdot 6 = 336 \quad (\text{correct}) \]

7. **Conclusion**:
Therefore, the value of \( n \) is:
\[ n = 8 \]

### Final Answer:
\[ n = 8 \]