Key Terms
 Binomial Coefficient – The coefficient of a given term in a binomial expansion.
 The coefficient of the kterm in the expansion of is denoted .
 Binomial Expansion – An expression that is equivalent to a binomial raised to a positive integer power, and that results from carrying out the multiplication indicated by the power.
 Combinations – Unique selections of items, no repeats.
 Ex. If you have 3 different fruits: apple, pear, and banana, and must choose only 2, you wouldn’t choose [apple and pear], then [pear and apple] because those would be repeats. The combination of apple and pear will only happen once, no matter the order.
 Choosing 2, you can also have a pear with banana or an apple with banana.
 Ex. If you have 3 different fruits: apple, pear, and banana, and must choose only 2, you wouldn’t choose [apple and pear], then [pear and apple] because those would be repeats. The combination of apple and pear will only happen once, no matter the order.
 Binomial Theorem – A polynomial identity which states:
 Factorial – The product of n and all positive whole numbers less than n.
 The notation n! means “n factorial.”
 Ex. 4! = 4 • 3 • 2 • 1 = 24
 Ex. 8! = 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1 = 40,320
 There is a button on your calculator “n!” you can use!
 Pascal’s Triangle – A triangular arrangement of numbers in which each row gives you the coefficients of a binomial expansion.
 Any number in the triangle can be computed by taking the sum of the two adjacent numbers in the previous row.
 Permutation – A selection of objects in which the order of the objects matters.
 Use this to show how many ways can you arrange a certain number of items.
 0! = 1
Notes
 Pascal’s Triangle
 Do you see a diagonal pattern?

 1 + 1 = 2
 1 + 2 = 3
 1 + 3 = 4, so the coefficient of the 2nd term in the 4th line down will be 4.
 What will the 2nd term look like including the variables? Look for the pattern. Try writing it out on paper! Scroll ALL the way down to see the answer!
 Do you see a horizontal pattern (for the exponents)?
 From left to right, powers of x get smaller while powers of y get bigger
 The row will always start and end with the same power (which matches the row that it is in)
 The pattern found with Pascal’s triangle works for any binomial
 Using Pascal’s triangle may be faster than using the FOIL method when binomials have coefficients.
 Watch and copy the video below
 Ex.
 Look at the 3rd row of Pascal’s Triangle and replace ALL x’s with 2’s.
 Substitute 2 for x:
 Simplified:
 Ex.
 Rewrite as:
 Substitute 3x for x and substitute 2y for y:
 Simplify:
 Pascal’s Patterns Enumerated
 Expanding a binomial to the nth power gives n + 1 terms.
 The first term of will always be , and the last will always be .
 The sum of the exponents of each term of the expansion of is n.
 The exponents of the xterms in the expansion of will start at n and decrease by 1 for every consecutive term.
 The exponents of the yterms in the expansion of will start at zero and increase by 1 for every consecutive term.
 The coefficients follow the pattern of Pascal’s triangle for the nth row.
 Negative Values for Pascal’s Triangle Expansion
 For , raising a negative value to a power creates a pattern of alternating signs.
 For , if n is an even number, every sign will be positive; but, if n is an odd number, every sign will be negative.
 Ex.
 Pascal’s Triangle by Row (n) and Column (k)
 Rows: x
 Columns: y
 Permutations Using the Combination Formula
 is read “n choose k,”
 Formula can be used to find the kth term of the nth row of Pascal’s triangle
 It is defined by the quotient (ratio) of factorials
 And for some reason, 0! = 1
 Use a calculator to find the answers
 Ex. You can choose 3 out of 26 friends to go with you to the movies. How many possible combinations can you have?
 Since calculators can’t handle numbers that large without converting to scientific notation, here’s a shortcut!
Examples
 For
 Ex 1a. Which of the following could NOT be a term in the expansion?
 Answer: since each term’s exponents MUST add up to 6, “20xy” is the one that could NOT be a term in the expansion.
 Ex 1b. What is the expansion?
 Ex 1a. Which of the following could NOT be a term in the expansion?
 Ex 2. Calculate the 5th term for
 The first term will have 8 as the xexponent and zero as the yexponent.
 The fifth term will have 8−4 (so = 4) as the xexponent and 4 as the yexponent.
 The term will be positive because is positive.
 The coefficient from the triangle is 70, so the fifth term is:
 Pascal’s Triangle: The Next Level
 Coefficients of Pascal’s Triangle