Solutions:

\( S=\{(x,y,z):x,y,z\in2{},\;x+2y+3z=42,\;x,y,z\ge 0\} \Rightarrow x+2y=42-3z \)

\( z \)	\( 0 \)	\( 1 \)	\( 2 \)	\( 3 \)	\( 4 \)	\( 5 \)	\( 6 \)	\( 7 \)	\( 8 \)	\( 9 \)	\( 10 \)	\( 11 \)	\( 12 \)	\( 13 \)	\( 14 \)
\( x+2y \)	\( 42 \)	\( 39 \)	\( 36 \)	\( 33 \)	\( 30 \)	\( 27 \)	\( 24 \)	\( 21 \)	\( 18 \)	\( 15 \)	\( 12 \)	\( 9 \)	\( 6 \)	\( 3 \)	\( 0 \)

Number of set of \( (x,y,z) \)
\( x+2y=42\to 0,1,2,\ldots,21\Rightarrow 22 \) cases
\( x+2y=39\to 0,1,2,\ldots,19\Rightarrow 20 \)
\( x+2y=36\to 0,1,2,\ldots,18\Rightarrow 19 \)
\( x+2y=33\to 0,1,2,\ldots,16\Rightarrow 17 \)
\( x+2y=30\to 0,1,2,\ldots,15\Rightarrow 16 \)
\( x+2y=27\to 0,1,2,\ldots,13\Rightarrow 14 \)
\( x+2y=24\to 0,1,2,\ldots,12\Rightarrow 13 \)
\( x+2y=21\to 0,1,2,\ldots,10\Rightarrow 11 \)
\( x+2y=18\to 0,1,2,\ldots,9\Rightarrow 10 \)
\( x+2y=15\to 0,1,2,\ldots,7\Rightarrow 8 \)
\( x+2y=12\to 0,1,2,\ldots,6\Rightarrow 7 \)
\( x+2y=9\to 0,1,2,\ldots,4\Rightarrow 5 \)
\( x+2y=6\to 0,1,2,3\Rightarrow 4 \)
\( x+2y=3\to 0,1\Rightarrow 2 \)
\( x+2y=0\to 0\Rightarrow 1 \)
Total \( =169 \)

JEE MainsJEE AdvanceBITSATJEE Mains | PYQ

JEE Mains JEE Advance BITSAT
JEE Mains | PYQ