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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

Let N be the number of ways of distributing 8 chocolates of...

Let N be the number of ways of distributing 8 chocolates of different brands among 3 children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of N .

Correct answer

24

Step-by-step solution

8→(1,2,5) or (1,3,4) Number of ways ∣8‾∣2‾∣5‾∣1‾×∣3‾+∣8‾∣1‾∣3‾∣4‾×∣3‾=(8×7×62+8×7×6×56)×6=56×6(3+5)⇒56×48=2688 Sum of digits =24\begin{aligned} & 8 \rightarrow(1,2,5) \text { or }(1,3,4) \\ & \text { Number of ways } \frac{\underline{|8}}{\underline{|2} \underline{|5} \underline{|1}} \times \underline{|3}+\frac{\underline{|8}}{\underline{|1} \underline{|3} \underline{|4}} \times \underline{|3} \\ & =\left(\frac{8 \times 7 \times 6}{2}+\frac{8 \times 7 \times 6 \times 5}{6}\right) \times 6 \\ & =56 \times 6(3+5) \Rightarrow 56 \times 48=2688 \\ & \text { Sum of digits }=24\end{aligned}
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