We explore combinations and permutations.
Firstly, we must establish the definition of n!, which just means n*(n-1)*(n-2)*...*1. Combinations are expressed as nCr, which means n choose r. This is choosing r objects or values from n objects or values. nCr is equal to n!/r!(n-r)!. Permutations are expressed as nPr, which is equal to n!/(n−r)!.
What does nPr equal?
n!/r!(n-r)!
What does nCr equal when n=9 and r=7?
36
Linear relationships describe situations where one quantity changes at a constant rate with respect to another. They can be represented by equations of the form y = mx + c, where m is the gradient (rate of change) and c is the initial value or y-intercept.
A ride-share company charges a $4 base fee and $2 per kilometre. If another company charges no base fee but $2.50 per kilometre, for what distance do both companies cost the same?
2x + 4 = 2.5x → 4 = 0.5x → x = 8 km
The points (2, 5) and (8, 17) lie on a straight line. Find the gradient and write the equation of the line in y = mx + c form.
m = (17 - 5)/(8 - 2) = 12/6 = 2 → y = 2x + 1
We collect, display, and analyse data to make informed decisions. Measures of centre such as the mean, median, and mode, along with measures of spread like range and interquartile range, help us describe data patterns effectively.
We interpret statistical graphs and summaries to compare groups and make conclusions. Outliers can significantly affect the mean and overall interpretation of data.
The data set {5, 7, 7, 10, 12, 13, 15} represents the number of goals scored by a team each match. Find the median and range. If a new game with 20 goals is added, how does the mean change?
Median = 10, Range = 15 - 5 = 10; New mean = (5+7+7+10+12+13+15+20)/8 = 89/8 = 11.125
The mean score of 5 students on a test is 12. One student scored 18. What is the mean of the remaining four students?
Total = 12 × 5 = 60 → Remaining total = 60 - 18 = 42 → Mean = 42 ÷ 4 = 10.5
We calculate the surface area and volume of 3D shapes to solve real-world problems, such as determining material usage or container capacity. Understanding units and converting between them is essential for accurate measurement.
A cylindrical water tank has a radius of 4 cm and a height of 10 cm. If the tank is half full, what is the volume of water inside?
Full V = πr²h = 3.14 × 16 × 10 = 502.4 cm³ → Half full = 502.4 ÷ 2 = 251.2 cm³
A cube-shaped gift box has side length 6 cm. If it is to be wrapped in paper, how much paper is needed? Then, find how much volume it can hold.
Surface area = 6 × 6² = 216 cm², Volume = 6³ = 216 cm³