Pascal's Triangle and ⁿC_r values

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With reference to the appearance of  ⁿC_r in the binomial expansion of  ( a + x )ⁿ , the n refers to the power and the r refers to the position of the term in question in the series expansion. The term's position count starts from zero. So the first term has r = 0, the second r = 1 etc.

ⁿC_r is often notated as (ⁿ_r)

Pascal's Triangle contains ⁿC_r

You may find it useful to remember that the second term of each row is the same as the row number of that row (shown in blue).

Pascal's Triangle (rows shifted to the left)

The n and r in the table are the same as the n and r in the symbol ⁿC_r . Note that row count starts at zero, so for example row 5 is actually the sixth row down.

The value of any  ⁿC_r  can be:

• looked up directly from Pascal's Triangle above using the values of n and r, or
• can be found using the ⁿC_r button an your calculator.

For example:

⁴C₂ = 6 ,                                ⁵C₂ = ⁵C₃ = 10 ,                                  ⁶C₀ = ⁶C₆ = 1

ⁿC_r is used when working out the coefficient of term number r (counting from zero), ie the x^r term, for the series expanson of  ( a + x )ⁿ .

EXAMPLE 1: What is the coefficient of the x³ term in the expansion of (a + x)⁵ ?

The x³ term is ⁵C₃a²x³ = 10a²x³, so the coefficient is 10a². (Note that the indices of a and x always add up to n - in this case 5.)

EXAMPLE 2: What is the coefficient of the third term, (ie term number 2 counting from zero - so r = 2) in the expansion of (a + x)⁶ ?

The r=2 term is ⁶C₂a⁴x² = 15a⁴x², so the coefficient is 15a⁴.