Question

The coefficient of \(a^{10}b^7c^3\) in the expansion of \((bc + ca + ab)^{10}\) is:

• A. 140
• B. 150
• C. 120
• D. 160

Hint:

When working with multinomial expansions, use the relationships between the powers of the terms and solve the system of equations to find the specific term.

Answer 1

The Correct Option is C

The given expression is:

\[ (bc + ca + ab)^{10}. \]

Each term in the expansion arises from the multinomial expansion of \((bc + ca + ab)^{10}\). Let:

\[ x = bc, \quad y = ca, \quad z = ab. \]

The multinomial expansion of \((x + y + z)^{10}\) is:

\[ (x + y + z)^{10} = \sum_{i+j+k=10} \frac{10!}{i!j!k!} x^i y^j z^k. \]

Substitute back \(x = bc, \; y = ca, \; z = ab\):

\[ x^i y^j z^k = (bc)^i (ca)^j (ab)^k = b^{i+k} c^{i+j} a^{j+k}. \]

Step 1: Match powers of \(a^{10}b^{7}c^{3}\).

We need:

\[ a^{10} \implies j + k = 10, \quad b^{7} \implies i + k = 7, \quad c^{3} \implies i + j = 3. \]

Solve these equations simultaneously:

1. \(j + k = 10\)
2. \(i + k = 7\)
3. \(i + j = 3\)

From \(i + j = 3\), substitute \(j = 3 - i\) into \(j + k = 10\):

\[ 3 - i + k = 10 \implies k = 7 + i. \]

Substitute \(k = 7 + i\) into \(i + k = 7\):

\[ i + (7 + i) = 7 \implies 2i + 7 = 7 \implies i = 0. \]

Using \(i = 0\), find \(j\) and \(k\):

\[ j = 3 - i = 3, \quad k = 7 + i = 7. \]

Step 2: Coefficient of the term.

The coefficient is given by the multinomial coefficient:

\[ \frac{10!}{i!j!k!} = \frac{10!}{0! \cdot 3! \cdot 7!}. \]

Simplify:

\[ \frac{10!}{3! \cdot 7!} = \frac{10 \cdot 9 \cdot 8}{3 \cdot 2 \cdot 1} = 120. \]

Conclusion:

The coefficient of \(a^{10}b^{7}c^{3}\) is:

\[ \boxed{120}. \]