4.1 Basic concepts of binomial theorem

Many problems in mathematics and engineering involve raising a two-term expression to a whole-number power, such as the growth factor (1 + r)ⁿ of a deposit or the cross-section term (x + y)³ in a design formula. The binomial theorem gives a fast and reliable way to deal with such powers without multiplying brackets repeatedly. Before the theorem itself can be used (it is developed in 4.2), you need three foundations: the language and notation of binomial expressions, the factorial, and the binomial coefficients that are built from factorials and found either by calculation or from Pascal's triangle.

4.1.1 Binomial expressions and notation.

A binomial expression is an algebraic expression made of exactly two terms joined by a plus or minus sign. The word comes from "bi" (two) and "nomial" (term). Binomials appear whenever a quantity has two parts, for example a base amount plus an increase, or a length plus an allowance.

Example: A SACCO pays interest of r per year on member deposits, so each year a deposit is multiplied by the binomial (1 + r). After 3 years at r = 0.08 the growth factor is (1 + 0.08)³.

Working:

(1.08)² = 1.08 × 1.08 = 1.1664

(1.08)³ = 1.1664 × 1.08 = 1.259712

A deposit of Ksh 50,000 becomes 50,000 × 1.259712 = Ksh 62,985.60

Key terms used with binomials

• Term: A part of an expression separated from the others by a plus or minus sign. In 3x + 7 the terms are 3x and 7.
• Coefficient: The numerical factor multiplying a variable. In 3x + 7 the coefficient of x is 3.
• Constant term: A term with no variable, such as 7 in 3x + 7.
• Index (exponent or power): The small raised number that shows how many times the base is multiplied by itself. In (a + b)ⁿ the index is n.
• Binomial power: A binomial raised to an index, such as (x + 2)⁵. The binomial theorem deals with powers where n is a non-negative whole number.
• Base: The expression that is being raised to the index. In (x + 2)⁵ the base is the binomial x + 2.

Binomials compared with other expressions

Each term of a binomial may itself be a number, a variable, a product or a fraction. The following are all binomials: x + 5, 3a − 2b, x² + 1/x and 4m + 7n².

Standard notation

The general binomial power is written (a + b)ⁿ, where a and b are the two terms and n is the index. The following points show how the notation is read and used:

• Index 0: (a + b)⁰ = 1, for any a and b where a + b ≠ 0.
• Index 1: (a + b)¹ = a + b.
• Index 2: (a + b)² means (a + b)(a + b), the binomial multiplied by itself.
• Index n: (a + b)ⁿ means n identical factors of (a + b) multiplied together.
• Subtraction: A difference is treated as a sum with a negative term, so (a − b)ⁿ = (a + (−b))ⁿ. The minus sign belongs to the term b and must be carried through every calculation.
• Compound terms: In (2x + 3y)⁴ the first term is a = 2x and the second term is b = 3y, so the coefficients and powers inside each term are part of a and b.

4.1.2 Factorials and their use in binomial coefficients.

The factorial of a non-negative whole number n, written n!, is the product of all whole numbers from n down to 1. Factorials count the number of ways of arranging n different objects in a row, and they form the building blocks of every binomial coefficient.

Definition:

n! = n × (n − 1) × (n − 2) × ... × 3 × 2 × 1

Example: A hotel has 4 different conference groups and 4 rooms, one group per room. The number of ways of allocating the rooms is 4!.

4! = 4 × 3 × 2 × 1 = 24 ways

Values of small factorials

Properties of factorials

• Recursive rule: n! = n × (n − 1)!. For example, 6! = 6 × 5! = 6 × 120 = 720. This rule lets you stop expanding a factorial as soon as a smaller factorial appears.
• Zero factorial: 0! = 1 by definition. This is consistent with the recursive rule, because 1! = 1 × 0! forces 0! = 1.
• Defined only for whole numbers: Factorials of negative numbers or fractions are not used in this topic.
• Rapid growth: Factorials grow very quickly, so 10! is already over three million. Always cancel before multiplying.
• No addition or subtraction rule: (a + b)! is not a! + b!, and (a − b)! is not a! − b!. For example, 5! − 3! = 120 − 6 = 114, which is not 2! = 2.

Simplifying fractions of factorials

Binomial coefficients are fractions of factorials, so you must be able to simplify them without a calculator. Follow these steps:

1. Write the fraction and identify the largest factorial in the numerator.
2. Expand that factorial only far enough to reach the largest factorial in the denominator, using n! = n × (n − 1)!.
3. Cancel the common factorial in the numerator and denominator.
4. Expand any remaining small factorials in the denominator into their products.
5. Cancel common numerical factors between the numerator and the denominator.
6. Multiply the remaining factors to obtain the result.
7. Check that the answer is a whole number, since binomial coefficients are always whole numbers.

Example: Simplify 7!/5!

7!/5! = (7 × 6 × 5!)/5! = 7 × 6 = 42

Example: Simplify 9!/(4! × 5!)

9!/(4! × 5!) = (9 × 8 × 7 × 6 × 5!)/(4! × 5!)

= (9 × 8 × 7 × 6)/(4 × 3 × 2 × 1)

= 3 024/24

= 126

Link to binomial coefficients

The factorial is the key ingredient in the formula for a binomial coefficient: nCr = n!/(r! × (n − r)!). The numerator counts all arrangements of n objects, and the factorials in the denominator remove the arrangements that are not distinct when the order of the chosen objects does not matter. The next sections show how to use this formula.

4.1.3 Binomial Coefficients

Binomial coefficients are the whole numbers that multiply the terms when a binomial power is expanded. They are written nCr (read "n choose r"), or as C(n, r), or as n above r in large brackets. Each coefficient can be found in two ways: by calculation using the combination formula, or by reading it from Pascal's triangle, and both methods are covered below.

4.1.3.1 Definition and calculation using combinations (nCr).

A combination is a selection of r objects from n different objects in which the order of selection does not matter. The binomial coefficient nCr is the number of such selections, and it is calculated with factorials.

Definition:

nCr = n!/(r! × (n − r)!), where n and r are whole numbers and 0 ≤ r ≤ n

Because (n − r)! cancels part of n!, a shorter form is often easier:

nCr = [n × (n − 1) × ... × (n − r + 1)]/r!

This form has exactly r factors in the numerator and r factors in the denominator.

Example: A county office must pick 3 officers from 8 volunteers to form a records committee. The number of possible committees is 8C3.

Formula: nCr = n × (n − 1) × ... /r!

Substitution: 8C3 = (8 × 7 × 6)/(3 × 2 × 1)

Working: 336/6 = 56

Result: 56 different committees

Example: A school chooses 3 prefects from 6 candidates.

6C3 = (6 × 5 × 4)/(3 × 2 × 1) = 120/6 = 20 different selections

Example: A farm wants to test a new fertiliser on 2 of its 7 plots.

7C2 = (7 × 6)/(2 × 1) = 42/2 = 21 possible choices of plots

Properties of binomial coefficients

• Choosing none: nC0 = 1, because there is only one way to choose nothing. Check: 9C0 = 9!/(0! × 9!) = 1.
• Choosing all: nCn = 1, because there is only one way to choose every object. Check: 5C5 = 1.
• Choosing one: nC1 = n. For example, 12C1 = 12.
• Symmetry: nCr = nC(n − r), because choosing r objects to take is the same as choosing n − r objects to leave. For example, 7C5 = 7C2 = (7 × 6)/2 = 21. Use the smaller of r and n − r to reduce work.
• Whole numbers: Every nCr is a positive whole number, so a fractional answer shows an arithmetic error.
• Addition rule: nCr + nC(r + 1) = (n + 1)C(r + 1). For example, 4C1 + 4C2 = 4 + 6 = 10 = 5C2. This rule is the basis of Pascal's triangle.

Procedure for calculating nCr

1. Identify n (the total number of objects) and r (the number chosen).
2. If r is more than half of n, replace r by n − r using the symmetry property.
3. Write r factors in the numerator, starting at n and decreasing by 1 each time.
4. Write r! in the denominator as r × (r − 1) × ... × 1.
5. Cancel common factors between numerator and denominator.
6. Multiply the remaining numerator factors and divide by the remaining denominator factors.
7. State the result as a whole number.

Example: Calculate 10C7.

Symmetry: 10C7 = 10C3

Formula: 10C3 = (10 × 9 × 8)/(3 × 2 × 1)

Working: 720/6 = 120

Result: 10C7 = 120

On a scientific calculator, enter n, press the nCr key, then enter r and press equals. Use the calculator to check your hand working, not to replace it.

4.1.3.2 Pascal’s Triangle as a tool for finding binomial coefficients.

Pascal's triangle is a triangular arrangement of numbers in which every entry is a binomial coefficient. It lets you list all the coefficients for a given index without any factorial calculation, which is quick for small indices. Rows are numbered from row 0 at the top, and entries in each row are numbered from position 0 at the left, so the entry in row n at position r is nCr.

The triangle up to row 8

• Row 0: 1
• Row 1: 1 1
• Row 2: 1 2 1
• Row 3: 1 3 3 1
• Row 4: 1 4 6 4 1
• Row 5: 1 5 10 10 5 1
• Row 6: 1 6 15 20 15 6 1
• Row 7: 1 7 21 35 35 21 7 1
• Row 8: 1 8 28 56 70 56 28 8 1

Procedure for building the triangle

1. Write 1 in row 0.
2. Start each new row with 1 at the far left.
3. For each position in the middle of the row, add the two numbers directly above it, one to the upper left and one to the upper right.
4. End the row with 1 at the far right.
5. Check that the row has one more entry than the row above it, so row n has n + 1 entries.
6. Check that the row is symmetrical, reading the same from left and right.
7. Repeat until you reach the row equal to the index n you need.

Example: Build row 6 from row 5 (1, 5, 10, 10, 5, 1).

1 (start)

1 + 5 = 6

5 + 10 = 15

10 + 10 = 20

10 + 5 = 15

5 + 1 = 6

1 (end)

Row 6 is 1, 6, 15, 20, 15, 6, 1.

Why the triangle works

Each entry is the sum of the two above it because of the addition rule: nCr = (n − 1)C(r − 1) + (n − 1)Cr. For example, 6C2 = 5C1 + 5C2 = 5 + 10 = 15, which matches the entry in row 6.

Reading coefficients from the triangle

Example: A contractor must check 2 concrete cube samples chosen from 6 cubes cast on one day. The number of possible pairs is 6C2. In row 6, position 2 (counting from 0) is 15, so there are 15 possible pairs.

Example: A hospital ward needs to pick 2 nurses from 5 for a night shift. The number of choices is 5C2, which is the entry at position 2 in row 5, so there are 10 choices.

Useful patterns in the triangle

• Outer edges: The first and last entries of every row are 1, matching nC0 = 1 and nCn = 1.
• Second entry: The second entry of row n is n, matching nC1 = n.
• Symmetry: Each row reads the same in both directions, matching nCr = nC(n − r).
• Row sums: The entries of row n add up to 2ⁿ. For row 5: 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2⁵.
• Limitation: To reach row n you must build every row above it, so for large n, such as 20C8, the factorial formula is quicker.