Difference between revisions of "Pascal Triangle Related Problems"
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− | 1. You need to find the 6th number (remember the first | + | 1. You need to find the 6th number (remember the first number in each row is considered the 0th number) of the 10th row in Pascal's triangle. |
The 10th row is: | The 10th row is: |
Revision as of 00:01, 4 May 2007
The triangle
Here are lines zero through twelve of Pascal's triangle:
Row 0: 1 Row 1: 1 1 Row 2: 1 2 1 Row 3: 1 3 3 1 Row 4: 1 4 6 4 1 Row 5: 1 5 10 10 5 1 Row 6: 1 6 15 20 15 6 1 Row 7: 1 7 21 35 35 21 7 1 Row 8: 1 8 28 56 70 56 28 8 1 Row 9: 1 9 36 84 126 126 84 36 9 1 Row 10: 1 10 45 120 210 252 210 120 45 10 1 Row 11: 1 11 55 165 330 462 462 330 165 55 11 1 Row 12: 1 12 66 220 495 792 924 792 495 220 66 12 1
Problem 1
You are given the binomial . Raise it to the 10th power.
Find:
1. The coefficient of the term.
2. The sum of the coefficients.
Solution
1. You need to find the 6th number (remember the first number in each row is considered the 0th number) of the 10th row in Pascal's triangle.
The 10th row is:
Row 10: 1 10 45 120 210 252 210 120 45 10 1
Thus the coefficient is the 6th number in the row or .
This can also be found using the binomial theorem:
Through the summation, the binomial theorem will provide you with the coefficient if each term of the result. In our particular case, we are only looking for the coefficient of the term.
Since you are looking for term in , then and .
So the coefficient of the term is .
2. Since all the coefficients are found in the 10th row, we simply need to add the numbers in the 10th row together. This can be done by hand since there are relatively few numbers, but we could also use the following formula to sum up the numbers:
This summation formula simply adds up all the coefficients since gives us each of the coefficients. So, the sum is .
For your information, the final polynomial which results from is: