2006 iTest Problems/Problem 10
Find the number of elements in the first rows of Pascal's Triangle that are divisible by .
The pattern for rows of Pascal's Triangle with the multiples of colored red is here: http://www.catsindrag.co.uk/pascal/?r=64&m=4 There are five different figures in this triangle.
The black triangles with red dots in them. There are of these.
The three small red triangles with a dot in the middle separated by black in between. There are of these.
The three red dots with a red triangle in the middle separated by black in between. There are of these.
The medium red triangles. There are of these.
The large red triangles. There are of these.
For the first figure, there are multiples of represented by the three red dots.
For the second figure, notice the first one of those is on the th row, meaning there are total numbers in that row. Then subtract the black numbers to get multiples, but that's for both of those lines, so each one is numbers long. The number of red numbers in the row of the triangle below that one is numbers long and the last row has number. Each one of those triangles therefore has numbers. In each copy of this figure, there are three of these triangles and a single dot adding to numbers.
For the third figure, there is one of the smaller triangles from the previous figure and three dots adding to numbers.
For the fourth figure, notice the first one of these triangles is on the th row so there are 17 numbers in that row. Subtract three for numbers in total for the tops of those two triangles and for one of them. Once again, that means one triangle has on the first row, on the second, until on the last row. This adds to a total of Since each of these figures are only one triangle, there are numbers.
For the fifth figure, we use the same logic to find that each large triangle has numbers
Therefore, the total number of red numbers, or multiples of four, are:
Instead of spitting the problem into five different figures, we could have noted that there are red dots, small triangles, medium triangles, and big triangles. Then, using similar logic to above, we find there are dots per small triangle, dots per medium triangle, and dots per large triangle. Then the answer is