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.
1,200 fundamentals
“A leaking bucket losing half its water each minute has total water = a/(1−½).”
An infinite geometric series sums to S∞ = a/(1−r) only if |r| < 1. If |r| ≥ 1 the terms do not shrink enough and the series diverges. Forgetting the |r| < 1 condition yields a false finite answer.
S∞ = a/(1−r)
Memory trick: Converge only when |r|<1 – ‘less than one’ keeps shrinking
“Walking up a staircase adds a fixed height each step, while a car that doubles speed each hour multiplies – AP vs GP.”
In an AP each term increases by a fixed addition d; in a GP each term increases by a fixed multiplication r. The pattern of constant differences versus constant ratios distinguishes them. Using the wrong formula is a typical trap.
Memory trick: Add → AP, Multiply → GP
“Adding the prices of 5 books gives a total, but an endless stream of coins adds up only if each coin’s value keeps shrinking.”
A finite series has a definite number of terms, so its sum is always calculable. An infinite series has a sum only when it converges, e.g., a GP with |r| < 1; otherwise it diverges. Writing S = a/(1−r) without checking convergence is wrong.
Memory trick: Finite = countable, Infinite = need convergence check
“When a ball drops from height a and rebounds to r·a each bounce, the total distance is a(1+r)/(1−r), which only works because |r|<1.”
An infinite geometric series Σ a rⁿ converges to a/(1−r) only when the common ratio satisfies |r|<1; otherwise the terms never vanish and the series diverges. The exam trap is using a/(1−r) without first checking the |r|<1 condition.
Memory trick: Check |r|<1 first, then apply a/(1−r).
“Arranging 3 books on a shelf gives a different order each time you shuffle them.”
nPr = n!/(n−r)! counts ordered arrangements of r items from n. Use when the sequence matters, like seat assignments. The common trap is applying nCr, which ignores order and gives a smaller count.
Memory trick: P = order, C = no order.
“Selecting 3 friends for a team, you don't care who stands left or right.”
nCr = n!/[r!(n−r)!] counts selections where order is irrelevant. It’s the right tool for committees or lottery tickets. Mistaking it for nPr overcounts by a factor of r!.
Memory trick: C = choose, ignore order.
“Picking a shirt (4 colors) and shoes (3 styles) makes 4×3 =12 distinct outfits.”
If one task has m ways and another independent task has n ways, total ways = m×n (Fundamental Counting Principle). The exam trap is adding when both choices must happen together.
Memory trick: AND → ×, OR → +.
“A 2-letter then 2-digit password uses 26×26×10×10 possibilities, not a sum.”
When choices are independent and must all occur, multiply their counts; adding applies only to mutually exclusive options. Forgetting this inflates or deflates the answer dramatically.
Memory trick: AND → ×, OR → +.
“How many 4-digit PINs without repeating digits? Use nPr with n=10, r=4.”
nPr = n!/(n−r)! gives ordered arrangements without repetition, useful for codes, seating, or arranging objects in a line. Using nCr here ignores order and yields far fewer possibilities.
nPr = n!/(n−r)!
Memory trick: P = n!/(n−r)!.
“Choosing 3 toppings from 10 for a pizza, the order you place them doesn’t matter, so you use nCr.”
nCr = n!/[r!(n−r)!] gives the number of ways to select r objects from n without regard to order. It is obtained by dividing the total permutations nPr by r! because each selection can be arranged in r! ways. Use it only when order is irrelevant.
nCr = n!/[r!(n−r)!]
Memory trick: nCr = nPr ÷ r! → divide permutations by internal order
“A 4-digit password (1234) changes if you swap digits, but picking 4 players for a team does not.”
Permutation counts ordered arrangements; combination counts unordered selections. nPr = n!/(n−r)! and nCr = n!/[r!(n−r)!] = nPr / r!. Use permutation when sequence matters, combination when it doesn’t.
Memory trick: P = C × r! → order = combination × internal swaps
“When setting a 4-digit PIN you can repeat digits, but assigning seats in a row to 4 distinct people you cannot repeat a person.”
With repetition each choice can be reused, giving n^r possibilities for r positions. Without repetition each item is used at most once, giving nPr = n!/(n−r)!. Pick the formula that matches whether reuse is allowed.
Memory trick: Rep → n^r; No rep → nPr
“Choosing 5 friends for a quiz team is like picking a committee – who sits where doesn’t matter.”
When only the selected objects matter and their arrangement does not, count with combinations C(n,r)=n!/(r!(n-r)!). Using P(n,r) treats each ordering as different and inflates the answer. Look for words like ‘committee’, ‘group’, ‘select’ to decide on C; ‘arrange’, ‘line up’, ‘form a number’ signal P.
Memory trick: C = Choose, P = Position – if no position, pick C.
“Opening a chocolate box with a rows of dark and b rows of milk layers, the total ways to pick a piece follow Pascal’s triangle.”
(a+b)ⁿ expands as Σ_(r=0)ⁿ C(n,r) a^(n-r) b^r, where the coefficients are the entries of Pascal’s triangle. The index r starts at 0, giving n+1 terms. Mixing up the start index shifts every term’s coefficient.
Memory trick: r=0→first term, total terms=n+1
“Mixing red and blue paint, each shade’s exponent must add up to the total number of strokes n.”
In every term of (a+b)ⁿ the exponents of a and b always sum to n: a^(n-r) b^r. Missing this rule produces terms that do not belong to the expansion and leads to incorrect coefficients.
Memory trick: a-exp + b-exp = n
“Finding the 5th gear ratio in a bike chain uses T5 = nCr a^(n-r) b^r with r=4.”
The (r+1)th term of (a+b)ⁿ is T_(r+1)=C(n,r) a^(n-r) b^r. To locate a specific term, set the required exponent, solve for r, then plug into the formula. Using r instead of r+1 shifts the term index.
Memory trick: T_(r+1)=C(n,r) a^(n-r) b^r
“Counting seats in a theater row, the 7th seat corresponds to r=6, not r=7, showing the +1 index rule.”
The binomial formula gives the (r+1)th term, not the rth. After finding r from the exponent condition, add 1 to obtain the term’s position. Forgetting the +1 yields a wrong term number and coefficient.
Memory trick: Formula → (r+1)th term
“In a 7-layer cake the central slice is the 4th term; with 8 layers there are two central slices (4th and 5th).”
Number of terms = n+1. If n is even, there is a single middle term at r = n/2. If n is odd, two middle terms occur at r = (n-1)/2 and r = (n+1)/2. Mis-counting n’s parity gives the wrong middle term(s).
Memory trick: Even n → one middle; odd n → two middles
“When you split a chocolate bar of 7 pieces (n=6), one piece sits exactly in the center; with 8 pieces (n=7) two equal pieces share the center.”
In (a+b)ⁿ the expansion has n+1 terms. If n is even, the (n/2+1)th term is the single middle term. If n is odd, the (n+1)/2th and (n+3)/2th terms are both middle. Forgetting the parity gives a wrong coefficient.
Memory trick: Even n → 1 middle; odd n → 2 middles
“Finding the (r+1)th seat in a theater row of n+1 seats uses T_(r+1)=nCr·a^(n−r)·b^r.”
The (r+1)th term of (a+b)ⁿ is nCr·a^(n−r)·b^r, where r runs from 0 to n. The exponent of a decreases while that of b increases by one each step. Using r outside 0-n makes the term vanish.
T_(r+1) = nCr·a^(n−r)·b^r
Memory trick: r starts at 0, term index = r+1
“In a kitchen, nCr tells how many ways to pick r spices, while the term is the actual mix of those spices raised to powers.”
The binomial coefficient nCr is only the numerical factor that multiplies the powers of a and b. The full term also contains a^(n−r)·b^r. Confusing the coefficient with the whole term drops the variable part and yields an incorrect expansion.
Memory trick: Coeff = count; term = count × powers
“Labeling pages from 1, page r is not the rth term; the term formula gives T_(r+1) for page r+1.”
The general term is denoted T_(r+1), meaning it is the (r+1)th term of the expansion. To find a specific term, set r = term number −1. Skipping this shift causes an off-by-one error in the answer.
Memory trick: Term number = r+1
“Drawing a unit circle, any point (cosθ, sinθ) lies exactly 1 unit from the origin.”
For any angle θ, sin²θ + cos²θ = 1 – the Pythagorean identity on the unit circle. The common mistake is writing sinθ + cosθ = 1; only the squares add to 1.
Memory trick: Squares, not sum, equal 1.
“The arc length of a circle segment s = r·θ works only when θ is measured in radians, not degrees.”
Radians and degrees are related by π rad = 180°. Calculus formulas (derivatives, limits, series) assume radian measure; using degrees introduces a factor π/180. Always convert to radians before applying.
Memory trick: Calc → rad, not deg.