# How many different triangles are possible

You have an unlimited supply of five different coloured pop-sticks, and want to make as many different coloured equilateral triangles as possible, using three sticks. One example is shown here. Two triangles are not considered different if they are rotations or reflections of each other.

How many different triangles are possible? ## 1 Answer

You can use 3 sticks of the same colour: 5 triangles

You can use 2 sticks of one colour, then one stick of another color: $5\times4=20$

You can also use 3 different colours of sticks. $5\times4\times3÷3=20$

Thus there are 45 possibilities.

• Please look into the question. It must be different coloured triangles – geetha Jul 10 '18 at 6:58
• How did you arrive 5 (different colors) * 4 (what is this number)? and the same applies for 3 different colors as well. – geetha Jul 10 '18 at 7:20
• @geetha What the question means is that two triangles are distinguished by the colours of the sides (which colours are used and how many sides are of a given colour). For the $2$ sticks of one colour and one stick of another colour case, there are $5$ ways of selecting the colour which is used on two sides, which leaves $4$ ways of selecting the colour of the other side. – N. F. Taussig Jul 10 '18 at 9:00