Binomial expansions are fundamental in various professional fields in Kenya, including finance, agriculture, healthcare, and information technology. Understanding how to expand binomial expressions allows professionals to simplify complex algebraic expressions encountered in budgeting, data analysis, and scientific computations. This chapter introduces the basic concepts of the binomial theorem, enabling students to apply these principles effectively in diverse workplace scenarios. Mastery of factorials and binomial coefficients is essential for accurate calculations and problem-solving.

5.1 Basic concepts of binomial theorem

The binomial theorem gives a method of expanding expressions of the form (a + b)ⁿ without multiplying the bracket by itself again and again. Electrical engineers meet such expressions when a quantity that depends on a power, such as power in a resistor (I²R) or motor load against speed (N³), changes by a small amount. This section covers the notation of binomial expressions, the factorials used to calculate coefficients, and the binomial coefficients themselves, including Pascal's triangle.

5.1.1 Binomial expressions and notation

A binomial expression has exactly two terms joined by a plus or minus sign, and the whole expression is raised to a power. Example: a resistor carrying a current I that rises by a small amount ΔI dissipates power P = (I + ΔI)² × R, and (I + ΔI)² is a binomial expression with n = 2.

Definition of binomial expressions

• Two terms only: A binomial consists of exactly two distinct algebraic terms separated by a plus or minus sign, for example (V + v) or (R − r).
• Raised to a power: The whole binomial is raised to a positive integer exponent n, which means the bracket is multiplied by itself n times.
• General form: It is written (a + b)ⁿ or (a − b)ⁿ, where a and b are numbers, variables or algebraic expressions.
• Use in modelling: Binomials model quantities in which a nominal value and a small change act together through a power law.
• Conductor area: The cross-sectional area of a circular conductor is proportional to d². If the diameter is d + Δd, the area is proportional to (d + Δd)².

Notation conventions

• Exponent notation: The power n is a non-negative integer. It tells you how many times the binomial multiplies itself.
• Terms inside parentheses: Brackets show that the exponent applies to the whole binomial. (a + b)² is not the same as a² + b².
• Variable representation: a and b may be plain numbers, symbols such as I and ΔI, or expressions such as 2x and 3y. When a term has a coefficient or a power, bracket it before expanding.
• Positive and negative signs: In (a − b)ⁿ the signs of the expanded terms alternate between + and −. In (a + b)ⁿ every term is positive when a and b are positive.
• Compact form: The notation lets you write a long product such as (a + b)(a + b)(a + b)(a + b)(a + b) as (a + b)⁵ before expanding it.

Examples of binomial expressions in electrical work

• Coil inductance: The inductance of an air-cored coil is proportional to N². Adding one turn gives a factor of (N + 1)².
• Motor load: The power taken by a fan or pump varies roughly as the cube of its speed, so a speed change gives (N + ΔN)³.
• Resistor tolerance: A resistor of nominal value R and fractional tolerance δ has a value R(1 + δ). Powers of (1 + δ) appear when the value is used in a squared or cubed relationship.
• Voltage deviation: A supply of V − ΔV volts supplying a heating element gives power proportional to (V − ΔV)².
• Conductor diameter: A drawn conductor whose diameter changes from d to d − Δd has an area proportional to (d − Δd)², a binomial with a negative second term.

Importance of understanding binomial notation

• Simplifies calculations: It allows an expansion to be written down without long manual multiplication.
• Prepares for coefficients: It leads directly to binomial coefficients and Pascal's triangle.
• Reduces errors: Clear bracket and exponent notation prevents mistakes such as squaring only one term.
• Facilitates communication: Standard notation lets engineers share working and check each other's calculations.
• Supports advanced topics: It is the base for approximations, series and probability work later in the course.

5.1.2 Factorials and their use in binomial coefficients

A factorial is the product of all positive whole numbers from 1 up to a given number n, written n!. Factorials count the ways of arranging objects, and they are the building blocks of every binomial coefficient. Example: the three phase conductors L1, L2 and L3 can be connected to the three terminals of a motor in 3! = 3 × 2 × 1 = 6 different orders, three of which give one phase sequence and three the reverse.

Definition and calculation of factorials

• Definition: The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n.
• Base case: By definition, 0! = 1. This makes the binomial coefficient formula work at the ends of a row, so that the first and last coefficients of an expansion come out as 1.
• Example: 5! = 5 × 4 × 3 × 2 × 1 = 120.
• Growth rate: Factorials grow very quickly. 6! = 720 and 10! = 3 628 800, and large factorials overflow a calculator display, so cancel common factors before multiplying.
• Cancelling in fractions: In a ratio, cancel the common factors first. 8! ÷ 6! = (8 × 7 × 6!) ÷ 6! = 8 × 7 = 56.

Properties of factorials

• Recursive nature: n! = n × (n − 1)!, so 7! = 7 × 6!. This is the quickest way to cancel in fractions.
• Multiplicative identity: 1! = 1, which is the first step of the recursive pattern.
• Relation to combinations: Every binomial coefficient is a ratio of factorials, nCr = n! ÷ (r! × (n − r)!).
• Positive integers only: The factorial is defined here for whole numbers 0, 1, 2, 3, …, so n and r in a binomial coefficient must be whole numbers.
• Order of operations: n! applies to n alone, so 2 × 3! = 2 × 6 = 12, whereas (2 × 3)! = 6! = 720.

Factorials in practical scenarios

• Meter positions: Three different instruments (an ammeter, a voltmeter and a wattmeter) can be mounted in three positions on a test panel in 3! = 6 ways.
• Testing sequence: The 5 final circuits on a distribution board can be tested in 5! = 120 different orders.
• Panel assignment: Four technicians can be assigned to four different switchboards in 4! = 24 ways.
• Energising order: Six feeders can be switched on in 6! = 720 possible sequences, which is why a fixed switching schedule is used.
• Lamp layout: Four different luminaires can be placed in a row in 4! = 24 arrangements.

Computing factorials efficiently

1. Identify n: Read the integer whose factorial is needed, for example 6 in 6!.
2. Write the product: Expand n! as n × (n − 1) × (n − 2) × … × 1, using n! = n × (n − 1)!.
3. Cancel first: If n! appears in a fraction, cancel the common factors before multiplying anything.
4. Multiply sequentially: Multiply from the largest factor downwards, writing each partial product, for example 6 × 5 = 30, 30 × 4 = 120, 120 × 3 = 360, 360 × 2 = 720.
5. Check with a calculator: For large n use the n! key and compare it with a small case worked by hand.
6. Use the value: Substitute the factorials into the binomial coefficient formula.

5.1.3 Binomial Coefficients

Binomial coefficients are the numerical factors that multiply the terms in the expansion of (a + b)ⁿ. They also give the number of ways of choosing r items from n different items, and there are two methods of finding them: the factorial formula and Pascal's triangle. Example: the number of ways of choosing 2 resistors from a tray of 6 different resistors to wire in series is a binomial coefficient, 6C2 = 15.

5.1.3.1 Definition and calculation using combinations (nCr)

The binomial coefficient for choosing r items from n, written nCr (also ⁿCᵣ or C(n, r), read "n choose r"), is the number of selections of r items from n different items when the order does not matter.

Formula:

nCr = n! ÷ (r! × (n − r)!)

Key properties of nCr:

• Range: r takes whole values from 0 to n.
• Symmetry: nCr = nC(n − r), so 10C8 = 10C2. Always calculate with the smaller of r and n − r.
• End values: nC0 = nCn = 1 and nC1 = n.
• Order ignored: Choosing resistors A and B is the same selection as choosing B and A, which is why r! appears in the denominator.
• Link to expansion: The coefficients nC0, nC1, …, nCn are the coefficients of the n + 1 terms of (a + b)ⁿ.

Steps for calculating nCr:

1. Identify n and r: Read the total number of items n and the number chosen r. For 3 breakers chosen from 8, n = 8 and r = 3.
2. Apply symmetry: If r is more than half of n, replace r with n − r to shorten the working.
3. Write the formula: nCr = n! ÷ (r! × (n − r)!).
4. Substitute: Put the numbers in, for example 8C3 = 8! ÷ (3! × 5!).
5. Cancel the larger factorial: Expand n! only as far as (n − r)!, for example 8! = 8 × 7 × 6 × 5!, and cancel 5!.
6. Multiply and divide: Work out the numerator and divide by r!.
7. State the result: Write the value as a whole number, since a coefficient is always an integer.

Example: Find 6C2, the number of ways of choosing 2 of 6 labelled resistors, R1 to R6.

nCr = n! ÷ (r! × (n − r)!)

6C2 = 6! ÷ (2! × 4!)

6C2 = (6 × 5 × 4!) ÷ (2 × 1 × 4!) = 30 ÷ 2

6C2 = 15 different pairs

Example: A technician chooses 3 of 8 circuit breakers to inspect. Find 8C3.

8C3 = 8! ÷ (3! × 5!)

8C3 = (8 × 7 × 6) ÷ (3 × 2 × 1) = 336 ÷ 6

8C3 = 56 possible selections

Example: Find 10C8 using symmetry.

10C8 = 10C2 = (10 × 9) ÷ (2 × 1) = 90 ÷ 2

10C8 = 45

5.1.3.2 Pascal’s Triangle as a tool for finding binomial coefficients

Pascal's triangle is a triangular array of numbers in which each row lists the binomial coefficients nC0, nC1, …, nCn for one value of n. It lets you write down coefficients by simple addition, without any factorials, which is quick for small powers. Example: row 4 reads 1, 4, 6, 4, 1, so 4C2 = 6.

Rows 0 to 8 of the triangle:

n = 0                 1
n = 1               1   1
n = 2             1   2   1
n = 3           1   3   3   1
n = 4         1   4   6   4   1
n = 5       1   5  10  10   5   1
n = 6     1   6  15  20  15   6   1
n = 7   1   7  21  35  35  21   7   1
n = 8 1   8  28  56  70  56  28   8   1

Rules for building the triangle:

1. Start: Row 0 is the single number 1.
2. Edges: Every row begins and ends with 1, because nC0 = nCn = 1.
3. Inside numbers: Each inner number is the sum of the two numbers directly above it. For example, in row 5 the 10 is 4 + 6 from row 4.
4. Symmetry: Each row reads the same forwards and backwards, matching nCr = nC(n − r).
5. Reading a coefficient: Count rows from n = 0 and positions from r = 0. The entry in row n at position r is nCr.
6. Row sums: The numbers in row n add up to 2ⁿ. Row 4 gives 1 + 4 + 6 + 4 + 1 = 16 = 2⁴, which is a quick check that a row is correct.

The addition rule is Pascal's rule: nCr + nC(r + 1) = (n + 1)C(r + 1).

Example: Use the triangle to find 6C3 and check it with the formula.

From row 6: 1, 6, 15, 20, 15, 6, 1, so 6C3 = 20.

Check: 6C3 = 6! ÷ (3! × 3!) = 720 ÷ 36

6C3 = 20

Example: Build row 9 from row 8 (1, 8, 28, 56, 70, 56, 28, 8, 1).

Add neighbours: 1, 1 + 8 = 9, 8 + 28 = 36, 28 + 56 = 84, 56 + 70 = 126, then repeat by symmetry.

Row 9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1

The triangle is best for small n, up to about 8 or 10. For larger n, or when only one coefficient is needed, use the nCr formula, because building every earlier row is slow and a single slip carries through all the rows below.

Role of binomial coefficients in expansions

In each term of (a + b)ⁿ the coefficient multiplies a product of a power of a and a power of b, so the coefficient fixes the size of the term. When a quantity such as power varies as (1 + δ)ⁿ for a small tolerance δ, the first few coefficients of the row give a close estimate of the result. The full expansion is developed in the next topic.

Chapter Summary

This chapter introduced the basic concepts of the binomial theorem, starting with binomial expressions and the notation used to represent them. It explained the role of factorials in calculating binomial coefficients and detailed how to compute these coefficients using combinations, supported by Pascal's Triangle as a practical tool. The general form of the binomial expansion was presented, including expansions for powers \(n = 2, 3, 4\) and special cases such as expansions of \((1 + x)^n\), as well as negative and fractional powers using the binomial series. The chapter demonstrated how binomial expansions simplify algebraic expressions and assist in solving polynomial equations. It also explored engineering applications where binomial expansions facilitate problem-solving and approximation. Finally, methods for estimating values in engineering calculations were discussed, highlighting the practical utility of binomial expansions in real-world engineering contexts.