Question Video: Finding Binomial Coefficients

Use Pascal’s triangle to determine the coefficients of the terms that result from the expansion of (𝑥 + 𝑦)⁶.

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Video Transcript

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.

We’ve been told what method we should use to find the coefficients, and that’s Pascal’s triangle. And that means our first step will be to reproduce Pascal’s triangle. Pascal’s triangle helps us to expand binomials in the form 𝑎 plus 𝑏 all to the 𝑛th power. For the triangle, we start with 𝑛 equals zero. The tip of the triangle is just a value of one.

Moving on to 𝑛 equals one, we get the second row that is two values of one. For the third row, we have 𝑛 equals two with the values one, two, one. We get the two in the middle by adding the ones just above it. Our 𝑛 equals three row will follow the pattern one, three, three, one. The two three terms come from the values one plus two above them. When 𝑛 equals four, we have the row one, four, six, four, one. When 𝑛 equals five, we have the row one, five, 10, 10, five, one. And for the row 𝑛 equals six, we have one, six, 15, 20, 15, six, and one.

The binomial we’re expanding is 𝑥 plus 𝑦 all to the sixth power. Our 𝑛-value equals six. And therefore, we’ll have the coefficients one, six, 15, 20, six, and one.

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