Question

Find the number of natural number solutions of $x_1+x_2+x_3+3x_4=15$.

• A. 78
• B. 94
• C. 106
• D. 103
• E. 87

Answer 1

The correct option is B 94

$\begin{array}{cccccc}& X_1& X_2& X_3& 3X_4& =15\\ \text{Case 1:}& X_1& X_2& X_3& 3& =15\\ \text{Case 2:}& X_1& X_2& X_3& 6& =15\\ \text{Case 3:}& X_1& X_2& X_3& 9& =15\\ \text{Case 4:}& X_1& X_2& X_3& 12& =15\end{array}$

Case 1:
$X_1+X_2+X_3=12;$ Total number of natural number solution = ${}^{11}{C}_2=55$
Case 2:
$X_1+X_2+X_3=9;$ Total number of natural number solution = ${}^8{C}_2=28$
Case 3:
$X_1+X_2+X_3=6;$ Total number of natural number solution = ${}^5{C}_2=10$
Case 4:
$X_1+X_2+X_3=3;$ Total number of natural number solution = 1
Total number of natural number solution = 55 + 28 + 10 + 1 = 94