Combinations — nCr, Pascal's triangle: Foundations

This page assumes you know nothing and builds every symbol the parent note uses, one brick at a time. Read top to bottom: each idea is the floor the next one stands on.

1. Counting distinct things — the very first picture

Before any formula, we need the idea of a distinct item: a thing you can tell apart from every other thing. Three friends named Amit, Bina, Chetan are distinct — you can point to each one.

Why the topic needs this: the parent formula counts subsets of $n$ distinct items. If items could repeat or were identical, "how many ways to choose" would mean something different. So "distinct" is the ground floor.

2. The symbol $n$ — "how many I have"

We use a letter instead of a fixed number so one formula can talk about any size pile at once. That is the whole reason algebra uses letters: write it once, use it forever.

3. The symbol $r$ — "how many I take"

Why both letters? The parent note constantly writes things like $(r n)$ — "from $n$ , choose $r$ ." You must feel these two roles: $n$ is the pool, $r$ is the grab.

4. Order matters vs. order doesn't — the heart of everything

This is the single distinction the whole topic turns on, so look at the picture slowly.

Why the topic needs this: a permutation ( $^{n} P_{r}$ ) is the ordered count; a combination ( $(r n)$ ) is the unordered count. The parent page's central move — dividing by $r!$ — exists only to turn the first into the second.

Recall Curly braces

${}$ When we write a group with curly braces ${a, b}$ , the braces are a signal: "order inside me is irrelevant." So ${a, b}$ and ${b, a}$ are literally the same set.

5. The exclamation mark: factorial $n!$

The picture behind it — factorial counts arrangements. How many ways can you line up 3 distinct people? The first slot has 3 choices, the next has 2 left, the last has 1 left: $3 \times 2 \times 1 = 3! = 6.$

Why the topic needs it: the number of orderings of a group of $r$ items is exactly $r!$ . That is the "un-shuffle" number you divide by. See Factorials and 0! for the full engine.

6. Falling products — $n (n - 1) (n - 2) \dots$

The parent note writes $^{n} P_{r} = n (n - 1) (n - 2) \dots (n - r + 1)$ , " $r$ factors." Let's earn that.

The picture — filling $r$ slots from a shrinking pool: slot 1 has $n$ choices, slot 2 has $n - 1$ (one used up), and so on for $r$ slots. That product IS $^{n} P_{r}$ .

7. The choose symbol $(r n)$

The picture: it is a count of pockets, not line-ups. Whenever the answer must not change when you shuffle the chosen items, this is your symbol.

8. Sigma-free reading of Pascal's triangle notation

The parent draws numbers as "row $n$ , position $r$ ." Two conventions you must lock in:

Why zero-indexing? Because $(0 n) = 1$ must be the first entry of every row, and $(0 0) = 1$ is the lone top. Starting counts at 0 makes the edges of the triangle always $1$ , matching "choose nothing / choose everything." See Pascal's Triangle patterns.

9. The plus sign that stacks the triangle

Pascal's rule $(r n) = (r - 1 n - 1) + (r n - 1)$ uses ordinary addition, but the reason you add is the counting law "disjoint cases add up."

This same "cases add, complement subtracts" idea powers the parent's "at least one" example and all of Probability — counting outcomes.

Prerequisite map

Read it as a staircase: distinctness and the two letters $n, r$ come first; factorials (and $0!$ ) power permutations; permutations plus "order doesn't matter" give combinations; combinations laid out by row give Pascal's triangle; and the "cases add" idea gives Pascal's rule.

Equipment checklist

Test yourself — cover the right side and answer before revealing.

What does "distinct items" mean?

Objects you can tell apart; swapping their names changes which one you mean.

(r n)

, what is

n

and what is

r

n

= size of the pool you choose from;

r

= how many you take, with

0 \leq r \leq n

What is the one difference between a permutation and a combination?

A permutation counts ordered line-ups; a combination counts unordered groups (order irrelevant).

Compute

4!

4 \cdot 3 \cdot 2 \cdot 1 = 24

Why is

0! = 1

There is exactly one way to arrange nothing — the empty arrangement.

Write the falling-product form of

^{7} P_{3}

and its value.

7 \cdot 6 \cdot 5 = 210

(three factors, last is

7 - 3 + 1 = 5

Build

(r n)

from

^{n} P_{r}

(r n) = \frac{^{n} P _{r}}{r !} = \frac{n !}{r ! ( n - r )!}

In Pascal's triangle, what are the row and position of the entry

(2 5)

, and its value?

Row

n = 5

, position

r = 2

; value

10

Why do the two Pascal-rule terms get added, not multiplied?

They are disjoint, exhaustive cases (item IN vs OUT), and disjoint cases add.

What does

{a, b} = {b, a}

tell you?

Curly braces mean order inside is irrelevant — it's the same set.

Connections

Permutations — nPr — the ordered count that combinations divide down.
Factorials and 0! — the multiplication engine and why $0! = 1$ .
Binomial Theorem — where these $(r n)$ show up as coefficients.
Probability — counting outcomes — "cases add, complement subtracts" in action.
Pascal's Triangle patterns — what the row/position grid reveals.
Combinations — nCr, Pascal's triangle — the parent topic this page prepares you for.