How many paths spell STATISTICS using Pascal's triangle?
S
T T
A A A
T T T T
I I I
S S
T
I I
C C C
S S
Thanks in advance
To find the number of paths that spell "STATISTICS" using Pascal's triangle, we need to examine the pattern of the letters and determine the number of unique paths that lead to each letter. Let's break it down step by step:
1. Start with the letter 'S' at the top. This is the starting point of our path.
2. The letter 'T' appears twice in the word, so there are two possible paths that lead to it. Each path can be represented as the branching out from 'S'.
3. Next, the letter 'A' appears three times. Since 'A' has two paths leading to it ('T' has two paths connected to it), we will consider it as two separate 'A' letters. Now we have four paths to continue from.
4. 'T' appears four times, so there are four paths leading to it.
5. The letter 'I' appears three times. Similar to step 3, we have already established four paths up to this point. Therefore, each 'I' letter will have four paths leading to it.
6. The letter 'S' appears twice. Since there are four paths at this point, each 'S' letter will have four paths leading to it.
7. Continuing with 'T', it becomes two 'T' letters again. The four paths branch out again to connect with each 'T' letter.
8. Finally, the letter 'C' appears three times. By this point, there are four paths remaining from the previous step. Each 'C' letter will have four paths leading to it.
Summing up all the paths, we can find the total number of paths that spell "STATISTICS" as follows:
Number of paths = 2 (for 'T') x 2 (for 'A') x 4 (for 'T') x 4 (for 'I') x 4 (for 'S') x 4 (for 'T') x 4 (for 'I') x 4 (for 'C') x 4 (for 'S')
= 2 x 2 x 4 x 4 x 4 x 4 x 4 x 4 x 4
= 32,768
Therefore, there are 32,768 paths that spell "STATISTICS" using Pascal's triangle.
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To find the number of paths that spell "STATISTICS" using Pascal's triangle, we can consider each letter as a step in a path. Starting from the first row of Pascal's triangle, we move diagonally down and to the right for each letter 'S', and diagonally down and to the left for each letter 'T'.
Let's break down the steps:
1) Start with the first row of Pascal's triangle (1).
2) Move diagonally down and to the right for the first 'S', resulting in the second row of Pascal's triangle (1, 1).
3) Move diagonally down and to the left for the first 'T', resulting in the third row of Pascal's triangle (1, 2, 1).
4) Move diagonally down and to the right for the second 'S', resulting in the fourth row of Pascal's triangle (1, 3, 3, 1).
5) Move diagonally down and to the left for the second 'T', resulting in the fifth row of Pascal's triangle (1, 4, 6, 4, 1).
6) Move diagonally down and to the right for the first 'I', resulting in the sixth row of Pascal's triangle (1, 5, 10, 10, 5, 1).
7) Move diagonally down and to the left for each of the 'I's, resulting in the seventh row of Pascal's triangle (1, 6, 15, 20, 15, 6, 1).
8) Move diagonally down and to the right for each of the 'C's, resulting in the eighth row of Pascal's triangle (1, 7, 21, 35, 35, 21, 7, 1).
9) Move diagonally down and to the left for each of the 'S's, resulting in the ninth row of Pascal's triangle (1, 8, 28, 56, 70, 56, 28, 8, 1).
Therefore, using Pascal's triangle, there are 8 different paths that spell "STATISTICS" in total.