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Use Pascal’s triangle to determine the coefficients of the terms that result from the expansion of 𝑥 plus 𝑦 all to the sixth power.
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We’ve been told what method we should use to find the coefficients, and that’s Pascal’s triangle.
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And that means our first step will be to reproduce Pascal’s triangle.
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Pascal’s triangle helps us to expand binomials in the form 𝑎 plus 𝑏 all to the 𝑛th power.
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For the triangle, we start with 𝑛 equals zero.
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The tip of the triangle is just a value of one.
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Moving on to 𝑛 equals one, we get the second row that is two values of one.
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For the third row, we have 𝑛 equals two with the values one, two, one.
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We get the two in the middle by adding the ones just above it.
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Our 𝑛 equals three row will follow the pattern one, three, three, one.
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The two three terms come from the values one plus two above them.
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When 𝑛 equals four, we have the row one, four, six, four, one.
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When 𝑛 equals five, we have the row one, five, 10, 10, five, one.
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And for the row 𝑛 equals six, we have one, six, 15, 20, 15, six, and one.
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The binomial we’re expanding is 𝑥 plus 𝑦 all to the sixth power.
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Our 𝑛-value equals six.
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And therefore, we’ll have the coefficients one, six, 15, 20, six, and one.