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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

There are several tea cups in the kitchen, some with handle and the...

There are several tea cups in the kitchen, some with handle and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly 1200. What is the maximum possible number of cups in the kitchen?

Correct answer

29

Step-by-step solution

Let cups without handle equals to x&x \& cups with handle equals to yy ⇒xC2×yC3=1200=24×3×52x(x−1)2×y(y−1)(y−2)6=24×3×52x=25,y=4 and x=16,y=5x+y is maximum when x=25,y=4\begin{aligned} & \Rightarrow { }^x C_2 \times{ }^y C_3=1200=2^4 \times 3 \times 5^2 \\ & \frac{x(x-1)}{2} \times \frac{y(y-1)(y-2)}{6}=2^4 \times 3 \times 5^2 \\ & x=25, y=4 \text { and } x=16, y=5 \\ & x+y \text { is maximum when } x=25, y=4 \end{aligned} maximum possible cups equals to 29
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