### Brain Teasers

# 21 Blocks

One day, Drakar was studying for an algebra test when he noticed that his younger sibling made a pyramid of 21 blocks. Each row, from the bottom up, had one less block than the lower one.

"Hmm, I wonder how many different pyramids I can make. Now, I can only switch the blocks with other blocks on the row. So blocks in the fourth row can only be switched with other blocks from the fourth row. And I can switch blocks from as many rows I like, and count it as one different pyramid."

After thinking for a minute he took out a calculator and quickly figured out the answer. How many different pyramids can be made?

"Hmm, I wonder how many different pyramids I can make. Now, I can only switch the blocks with other blocks on the row. So blocks in the fourth row can only be switched with other blocks from the fourth row. And I can switch blocks from as many rows I like, and count it as one different pyramid."

After thinking for a minute he took out a calculator and quickly figured out the answer. How many different pyramids can be made?

### Hint

First, find the number of rows. Next, find the number of combinations each individual one has.### Answer

24,883,200 different pyramids.First of all, there are six rows. (1 + 2 + 3 + 4 + 5 + 6 = 21) Now, the first row, if we disposed of the rest, has only one combination. The second has two, and the third has six. From this we can figure out that each row has a number of combinations equal to the last row's times the number of blocks in the current one. So four has 24, five has 120, and six, 720.

Now, while six has 720, each one of the fifth row's combinations gives you 720 MORE rows. same with five to four, etc. So we do 720 * 120 * 24 * 6 * 2 * 1. This ends up as 24,883,200 different pyramids.

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