Let $\overline{A = \{(a,b,c):a,b,c ext{~} ext{are non-negative integers} ext{~}}$ and $a + b + 2c

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Let $\overline{A = \{(a,b,c):a,b,c	ext{~}	ext{are non-negative integers}	ext{~}}$ and $a + b + 2c = 22\}$.The $n(A\,)$ is equal to :

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Sol. [Key Concept: For each $c$ the value $a + b$ is fixed, giving $C(a + b + 1,1)$ solutions; sum $1 + 3 + 5 + dots + 23$.]$${c = 0 \Rightarrow a + b = 22 \rightarrow \ ^{22 + 2 - 1}C_{2 - 1} = \ ^{23}C_{1} }{c = 1 \Rightarrow a + b = 20 \rightarrow \ ^{20 + 2 - 1}C_{2 - 1} = \ ^{21}C_{1} }{\vdots \ \vdots \ \vdots }{c = 10 \Rightarrow a + b = 2\ \rightarrow \ ^{2 + 2 - 1}C_{2 - 1} = \ ^{3}C_{1} }{c = 11 \Rightarrow a + b = 0 \rightarrow \ ^{0 + 2 - 1}C_{2 - 1} = \ ^{1}C_{1} }$$Number of non-negative integral solutions$${\ ^{1}C_{1} + \ ^{3}C_{1} + \ ^{5}C_{1} + \ldots + \ ^{23}C_{1} }{1 + 3 + 5\ldots\ldots + 23 }{= 144 }$$

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