Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Permutation and Combination

Olympiad workbook · IOQM · Permutation and Combination

If N is the number of triangles of different shapes (i.e. not...

If N is the number of triangles of different shapes (i.e. not similar) whose angles are all integers (in degrees), what is N/100 ?

Correct answer

27

Step-by-step solution

x+y+z=180x1+y1+z1=177 Total =177+2C2⇒ Total =179×1782=179×89 Total =3!(αβγ)+3(ααβ)+ααα=6(αβγ)+3(ααβ)+1\begin{aligned} & x+y+z=180 \\ & x^1+y^1+z^1=177 \\ & \text { Total }={ }^{177+2} C_2 \Rightarrow \text { Total }=\frac{179 \times 178}{2}=179 \times 89 \\ & \text { Total }=3!(\alpha \beta \gamma)+3(\alpha \alpha \beta)+\alpha \alpha \alpha \\ & =6(\alpha \beta \gamma)+3(\alpha \alpha \beta)+1 \end{aligned} For ( ααα\alpha \alpha \alpha ), number of ways = 1 For ( ααβ\alpha \alpha \beta ), 2α+β=1772 \alpha+\beta=177, number of ways = (88 cases) For αβγ\alpha \beta \gamma, number of ways =179×89−3×88−16=\frac{179 \times 89-3 \times 88-1}{6} =15931−2656=2611\begin{aligned} & =\frac{15931-265}{6} \\ & =2611 \end{aligned} So total way 2611+88+1=27002611+88+1=2700 Ans. 27
Practice Permutation and Combination on Quantrex Academy →

More from Permutation and Combination

All Permutation and Combination questions Full Permutation and Combination list All Olympiad workbook PYQs