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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

Find the number of ordered triples (a, b, c) of positive integers...

Find the number of ordered triples (a,b,c)(a, b, c) of positive integers such that 30a+50b+70c≤34330 a+50 b+70 c \leq 343.

Correct answer

30

Step-by-step solution

30a+50b+70c≤343⇒3a+5b+7c≤34.3⇒3a+5b+7c≤34a,b,c∈N,a=1+p,b=1+q,c=1+r3p+5q+7r≤19 If r=0,q=0⇒p can take 7 values q=1⇒p can take 5 values q=2⇒p can take 4 values q=3⇒p can take 2 values ⇒18 values  If r=1 then 3p+5q≤12q=0⇒p can take 5 values q=1⇒p can take 3 values q=2⇒p can take 1 values ⇒9 values  If r=2 then 3p+5q≤5q=0⇒p can take 2 values q=1⇒p can take 1 value ⇒3 values  Total 18+9+3=30 values 30a+50b+70c≤343⇒3a+5b+7c≤34.3⇒3a+5b+7c≤34\begin{aligned} & 30 a+50 b+70 c \leq 343 \\ & \Rightarrow 3 a+5 b+7 c \leq 34.3 \\ & \Rightarrow 3 a+5 b+7 c \leq 34 \\ & a, b, c \in N, \quad a=1+p, \quad b=1+q, \quad c=1+r \\ & 3 p+5 q+7 r \leq 19 \\ & \text { If } r=0, \\ & q=0 \Rightarrow p \text { can take } 7 \text { values } \\ & q=1 \Rightarrow p \text { can take } 5 \text { values } \\ & q=2 \Rightarrow p \text { can take } 4 \text { values } \\ & q=3 \Rightarrow p \text { can take } 2 \text { values } \\ & \Rightarrow 18 \text { values } \\ & \text { If } r=1 \text { then } 3 p+5 q \leq 12 \\ & q=0 \Rightarrow p \text { can take } 5 \text { values } \\ & q=1 \Rightarrow p \text { can take } 3 \text { values } \\ & q=2 \Rightarrow p \text { can take } 1 \text { values } \\ & \quad \Rightarrow 9 \text { values } \\ & \text { If } r=2 \text { then } 3 p+5 q \leq 5 \\ & q=0 \Rightarrow p \text { can take } 2 \text { values } \\ & q=1 \Rightarrow p \text { can take } 1 \text { value } \\ & \quad \Rightarrow 3 \text { values } \\ & \text { Total } 18+9+3=30 \text { values } \\ & 30 a+50 b+70 c \leq 343 \\ & \Rightarrow 3 a+5 b+7 c \leq 34.3 \\ & \Rightarrow 3 a+5 b+7 c \leq 34\end{aligned}
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