Question

The number of ordered triplets of positive integers satisfying \(20\le x+y+z\le50\) is

• A. \({}^{5}C_3\)
• B. \({}^{19}C_3\)
• C. \({}^{50}C_3-{}^{19}C_3\)
• D. \({}^{69}C_3-{}^{19}C_3\)

Hint:

Ordered triplets counted via combinations on transformed non-negative variables.

Answer 1

The Correct Option is C

Step 1: For positive integers, put \(x'=x-1,\;y'=y-1,\;z'=z-1\ge0\). Then \[ x+y+z=n \Rightarrow x'+y'+z'=n-3. \] Number of ordered solutions for fixed sum \(n\) is \({}^{n-1}C_2\).
Step 2: Required range: \[ n=20\text{ to }50. \]
Step 3: Total: \[ {}^{50}C_3-{}^{19}C_3. \] This is option (C).