Coefficient of `a^3b^4c^5` in expansion of `(bc+ca+ab)^6`

Answer

To find the coefficient of \( a^3b^4c^5 \) in the expansion of \( (bc + ca + ab)^6 \), we can use the multinomial theorem. Here’s a step-by-step solution:

### Step 1: Identify the terms in the expansion
The expression \( (bc + ca + ab)^6 \) consists of three terms: \( bc \), \( ca \), and \( ab \). We need to find how many times we can choose these terms to get \( a^3b^4c^5 \).

### Step 2: Set up the equations
Let:
- \( p \) be the number of times we choose \( bc \),
- \( q \) be the number of times we choose \( ca \),
- \( r \) be the number of times we choose \( ab \).

From the expansion, we have:
- The total number of terms chosen: \( p + q + r = 6 \)
- The power of \( b \): \( p + r = 4 \)
- The power of \( c \): \( p + q = 5 \)
- The power of \( a \): \( q + r = 3 \)

### Step 3: Solve the equations
We can solve these equations step by step.

1. From \( p + r = 4 \) (Equation 1) and \( p + q = 5 \) (Equation 2), we can subtract Equation 1 from Equation 2:
\[ (p + q) - (p + r) = 5 - 4 \implies q - r = 1 \implies q = r + 1 \quad \text{(Equation 3)} \]

2. Substitute \( q \) from Equation 3 into \( q + r = 3 \) (Equation 4):
\[ (r + 1) + r = 3 \implies 2r + 1 = 3 \implies 2r = 2 \implies r = 1 \]

3. Now substitute \( r = 1 \) back into Equation 3:
\[ q = 1 + 1 = 2 \]

4. Finally, substitute \( q = 2 \) into Equation 1:
\[ p + 1 = 4 \implies p = 3 \]

So we have:
- \( p = 3 \)
- \( q = 2 \)
- \( r = 1 \)

### Step 4: Calculate the coefficient
Using the multinomial coefficient formula:
\[ \text{Coefficient} = \frac{n!}{p!q!r!} \]
where \( n = 6 \), \( p = 3 \), \( q = 2 \), and \( r = 1 \):
\[ \text{Coefficient} = \frac{6!}{3! \cdot 2! \cdot 1!} \]

Calculating the factorials:
\[ 6! = 720, \quad 3! = 6, \quad 2! = 2, \quad 1! = 1 \]

Substituting these values:
\[ \text{Coefficient} = \frac{720}{6 \cdot 2 \cdot 1} = \frac{720}{12} = 60 \]

### Final Answer
The coefficient of \( a^3b^4c^5 \) in the expansion of \( (bc + ca + ab)^6 \) is \( 60 \).

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