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Weekly Challenge 22: Combinations of Two

Stage: 4 and 5 Short Challenge Level: Challenge Level:1
See all short problems arranged by curriculum topic in the short problems collection

 
pascal triangle
Can you use this diagram to prove that the number of different pairs of objects which can be chosen from six objects, $^6C_2$, is $$1 + 2 + 3 + 4 + 5?$$
Generalise this to show that the number of ways of choosing pairs from $n$ objects is
 $$^nC_2 = 1 + 2 + ...+ (n-1) = \frac{1}{2}n (n - 1).$$ 
Did you know ... ?
The sum of the first $n$ whole numbers is called a triangle number because this sum can be represented geometrically by a triangular array of dots. The sum is easily found by working out the number of dots in the parallelogram formed by putting two triangular arrays side by side.