Fundamentals
The core facts every aspirant should own — each a titled nugget with a real-world story, the concept in plain words, and a memory trick. Works even when the internet doesn't.
300 fundamentals
“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”
If rearranging the same items doesn't create a new outcome, use combinations, not permutations.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
(a+b)^n expands as a sum of terms nCr·a^(n−r)·b^r, with coefficients from Pascal's triangle.
Memory trick: in each term the exponents of a and b sum to n.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
A frequent error is forgetting that the powers of a and b add up to n in every term. In reality, (a+b)^n expands as a sum of terms nCr·a^(n−r)·b^r, with coefficients from Pascal's triangle.
Memory trick: in each term the exponents of a and b sum to n.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
The (r+1)th term is T_(r+1) = nCr·a^(n−r)·b^r.
Memory trick: to find a specific term, set the exponent and solve for r, then plug in.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
A frequent error is treating the general term as the rth term instead of the (r+1)th. In reality, the (r+1)th term is T_(r+1) = nCr·a^(n−r)·b^r.
Memory trick: to find a specific term, set the exponent and solve for r, then plug in.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
For even n there is one middle term; for odd n there are two.
Memory trick: count terms as n+1; the middle depends on whether n is odd or even.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
A frequent error is assuming there's always exactly one middle term. In reality, for even n there is one middle term; for odd n there are two.
Memory trick: count terms as n+1; the middle depends on whether n is odd or even.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
The general (r+1)th term of (a+b)^n. Use it when 0 ≤ r ≤ n.
T_(r+1) = nCr·a^(n−r)·b^r
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
The binomial coefficient nCr is just the number multiplying a term; the term itself also includes the powers of a and b.
“Pascal's triangle hides inside every binomial expansion — and even in the odds of coin tosses.”
The general term T_(r+1) is the (r+1)th term — watch the index when solving for a specific term.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
A frequent error is misremembering it as sinθ + cosθ = 1. In reality, for any angle, sin²θ + cos²θ = 1.
Memory trick: it's the SQUARES that add to 1, from the unit circle.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
Angles can be measured in degrees or radians (π rad = 180°); calculus formulas assume radians.
Memory trick: always switch to radians before differentiating trig functions.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
A frequent error is using degrees in derivative/limit formulas. In reality, angles can be measured in degrees or radians (π rad = 180°); calculus formulas assume radians.
Memory trick: always switch to radians before differentiating trig functions.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
A frequent error is accepting an answer where sinθ > 1. In reality, sinθ and cosθ always lie between −1 and 1.
Memory trick: if a step gives sinθ = 1.4, you've made an error somewhere.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The fundamental Pythagorean identity. Use it when any real angle θ.
sin²θ + cos²θ = 1
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The sine addition formula. Use it when compound angles.
sin(A±B) = sinA cosB ± cosA sinB
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
Degrees split a circle into 360 parts; radians measure the arc length per unit radius (2π in a full circle) and are used in calculus.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The sine rule (a/sinA = b/sinB) suits two angles and a side; the cosine rule (c²=a²+b²−2ab·cosC) suits two sides and the included angle.
“GPS pinpoints you by solving triangles between satellites — trigonometry from space.”
The range of both sine and cosine is [−1, 1] — a value outside means a mistake.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
The steepness of a line, m = (y₂−y₁)/(x₂−x₁) = tanθ.
Memory trick: slope is 'rise over run' — change in y over change in x.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
A frequent error is inverting the formula to (x₂−x₁)/(y₂−y₁). In reality, the steepness of a line, m = (y₂−y₁)/(x₂−x₁) = tanθ.
Memory trick: slope is 'rise over run' — change in y over change in x.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
Memory trick: parallel → equal slopes; perpendicular → m₁·m₂ = −1.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
A frequent error is thinking perpendicular lines have equal slopes. In reality, parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.
Memory trick: parallel → equal slopes; perpendicular → m₁·m₂ = −1.
“Every 'rate' you plot — speed, price, growth — is a straight line's slope telling a story.”
Slope-intercept y=mx+c, point-slope y−y₁=m(x−x₁), and two-point forms describe the same line.
Memory trick: use point-slope when you have a point and a slope.