How many different paths will spell the word BINOMIAL in the following arrangement(moving diagonally downwards to the left or right)?
B
I I
N N N
O O O O
M M M
I I
A
L L
My Answer:
Starting from B using Pascal's Triangle
B
1 1
1 2 1
1 3 3 1
4 6 4
10 10
20
20 20
There are 40 different paths. Is this correct? Thanks for your help.
2*3*4*3*2*1*2
I get 288
Your answer is incorrect. To find the number of different paths that spell the word BINOMIAL, we need to count the number of paths from each B, I, N, O, M, A, L to the next letter in the word.
Starting from B:
- There are 2 paths from B to the next I.
- From each I, there are 3 paths (diagonally downwards to the left or right) to the next N.
- From each N, there are 4 paths to the next O.
- From each O, there are 3 paths to the next M.
- From each M, there are 2 paths to the next I.
- From each I, there are 3 paths to the next A.
- From each A, there is 1 path to the next L.
- From each L, there is 1 path to the last letter, which is also L.
To find the total number of different paths, we multiply the number of paths at each step:
2 * 3 * 4 * 3 * 2 * 3 * 1 * 1 = 432
So, there are 432 different paths that spell the word BINOMIAL in the given arrangement.
To find the number of different paths that can spell the word "BINOMIAL" in the given arrangement, we can use the concept of Pascal's Triangle.
1. Start by labeling the first row with the number 1.
2. Each subsequent row in Pascal's Triangle is generated by adding the two numbers directly above it in the previous row.
3. Create the triangle until you reach the desired number of rows needed to spell the word "BINOMIAL".
Let's go through the steps to create the Pascal's Triangle for this particular problem:
Row 1: 1
Row 2: 1 1
Row 3: 1 2 1
Row 4: 1 3 3 1
Row 5: 1 4 6 4 1
Row 6: 1 5 10 10 5 1
Row 7: 1 6 15 20 15 6 1
Row 8: 1 7 21 35 35 21 7 1
Row 9: 1 8 28 56 70 56 28 8 1
Row 10: 1 9 36 84 126 126 84 36 9 1
Now, let's count the number of paths using this triangle. Each path starts from the top letter 'B' and progresses diagonally downwards to the left or right until we reach the letter 'L' at the bottom.
From the triangle, we can see that there are 20 different paths from the top 'B' to the bottom 'L'.
Therefore, your answer of 40 different paths is incorrect. There are 20 different paths to spell the word "BINOMIAL" in the given arrangement.