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

$\begin{array}{cccccc}Cases& X_1& X_2& X_3& 3X_4& =15\\ Case1:& X_1& X_2& X_3& 3& =15\\ Case2:& X_1& X_2& X_3& 6& =15\\ Case3:& X_1& X_2& X_3& 9& =15\\ Case4:& X_1& X_2& X_3& 12& =15\end{array}$
Case 1:
X${}_1$ + X${}_2$ + X${}_3$ = 12; Total number of natural number solutions = ${}^{11}$C${}_2$= 55
Case 2:
X${}_1$ + X${}_2$ + X${}_3$ = 9; Total number of natural number solutions = ${}^8$C${}_2$ = 28
Case 3:
X${}_1$ + X${}_2$ + X${}_3$ = 6; Total number of natural number solutions = ${}^5$C${}_2$ = 10
Case 4:
X${}_1$ + X${}_2$ + X${}_3$ = 3; Total number of natural number solutions =1
Total number of natural number solutions = 55 +28 + 10 + 1 = 94