IPMAT Indore 2024 · Quantitative Ability · Question 34

Step 1 — state when {}^{n}{C}_{r} is defined. We need

n\ge 0, r\ge 0, r\le n

with n and r non-negative integers. Here n=17-x and r=3x+1, so x must be an integer.

Step 2 — write the three inequalities.

17-x\ge 0 \Rightarrow x\le 17

3x+1\ge 0 \Rightarrow x\ge -1/3 \Rightarrow x\ge 0 \left( x\text{integer}\right)

3x+1\le 17-x \Rightarrow 4x\le 16 \Rightarrow x\le 4

Step 3 — intersect. The binding constraints are 0\le x\le 4:

x\in \{0,1,2,3,4\}

Step 4 — verify each.

Answer: Option 1 — 5

Shortcut. Only one inequality does any work. r\ge 0 and n\ge 0 are nearly free; the binding condition is always

r\le n \Rightarrow 3x+1\le 17-x \Rightarrow x\le 4

Combined with x\ge 0 that is \{0,1,2,3,4\} — count 5.

Two habitual slips: forgetting that x=0 is allowed (giving 4, option d), and forgetting r\le n altogether (giving 18 values). Check the endpoints explicitly — here x=4 gives {}^{13}{C}_{13}=1, which is perfectly legitimate.