Question

The number of ordered triplets of positive integers which satisfy the inequality $20\le x+y+w\le 50$

• A. ${}^{50}{C}_3{-}^{19}{C}_3$
• B. ${}^{50}{C}_3{+}^{19}{C}_3$
• C. ${}^{50}{C}_4{-}^{19}{C}_3$
• D. $Noneofthese$

Answer 1

Answer: (A)

${}^{50}{C}_3{-}^{19}{C}_3$

Given inequality is,

$\Rightarrow 20\le x+y+w\le 50$

Therefore, required number of ordered triplets will be

$=$ Number of ways to distribute $20+$ Number of ways to distribute $21+$ ...... Number of ways to distribute $50$

${=}^{19}{C}_2{+}^{20}{C}_2+.......{+}^{49}{C}_2$

$={[}^{19}{C}_3{+}^{19}{C}_2{+}^{20}{C}_2+........{+}^{49}{C}_2]{-}^{19}{C}_3$

$=\left[{(}^{19}{C}_2{+}^{19}{C}_3\right){+}^{20}{C}_2+.......{+}^{49}{C}_2]{-}^{19}{C}_3$

$=\left[{(}^{20}{C}_3{+}^{20}{C}_2\right)+.......{+}^{49}{C}_2]{-}^{19}{C}_3$

${=}^{50}{C}_3{-}^{19}{C}_3$