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Calculate

Calculate. 4C0, 4C1, 4C2, 4C3 and 4C4 4C0=1, 4C1=4, 4C2=6, 4C3=4 and 4C4=1 Where have you seen this sequence of numbers before?. Pascal’s Triangle Revisited. 0C0 1C0 1C1 2C0 2C1 2C2 3C0 3C1 3C2 3C3 4C0 4C1 4C2 4C3 4C4 n th line nC0 nC1 nC2 etc… (starts on the 0 th line).

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Calculate

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  1. Calculate • 4C0, 4C1, 4C2, 4C3 and 4C4 • 4C0=1, 4C1=4, 4C2=6, 4C3=4 and 4C4=1 • Where have you seen this sequence of numbers before?

  2. Pascal’s Triangle Revisited • 0C0 • 1C0 1C1 • 2C0 2C1 2C2 • 3C0 3C1 3C2 3C3 • 4C0 4C1 4C2 4C3 4C4 • nth line nC0 nC1 nC2 etc… • (starts on the 0th line)

  3. Implication: Pascal’s Formula 0C0 1C0 1C1 2C0 2C1 2C2 3C0 3C1 3C2 3C3 4C0 4C1 4C2 4C3 4C4 Notice that each entry is equal to the sum of 2 entries from a proceeding row. For example: 4C2=3C1 + 3C2 This is known as Pascal’s Formula: nCr=n-1Cr-1 + n-1Cr

  4. Binomial Theorem Do you see a pattern in the coefficients?

  5. Pascal’s Triangle • 1 • 1 1 • 2 1 • 1 3 3 1 • 1 4 6 4 1 • 1 5 10 10 5 1 • 1 6 15 20 15 6 1 • nth line nC0 nC1 nC2 etc… • (starts on the 0th line)

  6. Example: Expand (x+y)6 • Try it yourself, use row 6 from Pascal’s Triangle (the seventh row counting from the top) or 6C0, 6C1 … 6C6 as the coefficients.

  7. Another Example HINT: expand (a+b)4 then sub in 2x for a and 1/y for b.

  8. Practice • Page 293 • Questions 1 to 12

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