S∞ = a/(1−r)

“Compound interest is a geometric progression quietly growing your money (or your loan).”

Sum of an infinite geometric series. Use it when only when |r| < 1.

S∞ = a/(1−r)

arithmetic progression vs geometric progression

“Compound interest is a geometric progression quietly growing your money (or your loan).”

An AP grows by adding a fixed number each step; a GP grows by multiplying by a fixed ratio each step.

finite series vs infinite series

“Compound interest is a geometric progression quietly growing your money (or your loan).”

A finite series always has a sum; an infinite series has one only if it converges (e.g. a GP with |r|<1).

Myth: Infinite GP always sums

“Compound interest is a geometric progression quietly growing your money (or your loan).”

The sum a/(1−r) exists only when |r|<1; otherwise the series diverges.

Permutation — common mistake

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

A frequent error is using permutations when order doesn't matter. In reality, an arrangement where order matters: nPr = n!/(n−r)!.

Memory trick: if rearranging counts as different, it's a permutation.

Combination — common mistake

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

A frequent error is using combinations when order actually matters. In reality, a selection where order does not matter: nCr = n!/[r!(n−r)!].

Memory trick: if only the group matters (not its order), it's a combination.

Fundamental principle of counting

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

If one task has m ways and another n ways, together they have m×n ways.

Memory trick: 'and' usually means multiply; 'or' usually means add.

Fundamental principle of counting — common mistake

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

A frequent error is adding when you should multiply independent choices. In reality, if one task has m ways and another n ways, together they have m×n ways.

Memory trick: 'and' usually means multiply; 'or' usually means add.

nPr = n!/(n−r)!

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

Number of ordered arrangements of r from n. Use it when order matters, no repetition.

nPr = n!/(n−r)!

nCr = n!/[r!(n−r)!]

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

Number of unordered selections of r from n. Use it when order does not matter.

nCr = n!/[r!(n−r)!]

permutation vs combination

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

A permutation counts ordered arrangements (a password); a combination counts unordered selections (a team) — nCr = nPr / r!.

with repetition vs without repetition

“A 4-digit PIN has 10,000 possibilities — that's permutations protecting your account.”

With repetition, items can be reused (like digits in a PIN); without repetition, each item is used at most once (like ranking people).

Myth: Using P when order is irrelevant

“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.

Binomial expansion

“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.

Binomial expansion — common mistake

“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.

General term

“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.

General term — common mistake

“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.

Middle term

“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.

Middle term — common mistake

“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.

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 general (r+1)th term of (a+b)^n. Use it when 0 ≤ r ≤ n.

T_(r+1) = nCr·a^(n−r)·b^r

binomial coefficient vs term

“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.

Myth: General term is the rth term

“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.

Pythagorean identity — common mistake

“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.

Radians vs degrees

“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.