General term of binomial expansion — finding specific terms: Foundations

Before you can read the parent note fluently, you need to own every symbol it fires at you. This page builds each one from absolute zero, in an order where each idea leans on the one before it. Nothing is assumed.

0. What a "term" and a "coefficient" even are

WHAT. A term is a single chunk in a sum that is separated by $+$ or $-$ signs. In $5 x^{3} + 7 x - 2$ there are three terms: $5 x^{3}$ , $7 x$ , and $- 2$ .

The picture. Look at the first figure: a sum is a train, and each term is one carriage. You can point at one carriage without touching the others.

Figure — General term of binomial expansion — finding specific terms

A coefficient is the plain number multiplying the variable part of a term. In $5 x^{3}$ the coefficient is $== 5 ==$ and the variable part is $x^{3}$ .

WHY the topic needs it. The whole parent topic is about grabbing one carriage from a very long train (the expansion) — either the carriage itself (a "specific term") or just the number painted on it (a "coefficient"). If those two words are fuzzy, the questions are unreadable.

1. Exponents (powers) — the little raised number

WHAT. $a^{n}$ means "multiply $a$ by itself $n$ times": $a^{4} = a \cdot a \cdot a \cdot a$ . The raised number $n$ is the exponent (or power, or index); $a$ is the base.

The picture. $2^{3}$ is a stack of three identical factors of $2$ — a tower three blocks tall. The exponent is the height of the tower.

Two rules the parent uses constantly:

Negative and zero exponents — these show up in Examples 2 and 3:

WHY the topic needs it. In $(x^{2} + \frac{1}{x})^{12}$ the two pieces are $x^{2}$ (power $+ 2$ ) and $x^{- 1}$ (power $- 1$ ). The whole method — "collect the power, set it equal, solve" — is nothing but adding these exponents using the rules above. And "term independent of $x$ " literally means "the exponent equals $0$ ," which needs $x^{0} = 1$ .

2. Factorials — the shrinking-product symbol $n!$

WHAT. $n!$ (read " $n$ factorial") means "multiply all whole numbers from $n$ down to $1$ ": $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120.$

The picture. In the second figure, $n!$ is a staircase of factors that shrinks by one at each step until it hits $1$ .

One special value you must accept:

WHY the topic needs it. The binomial coefficient $(r n)$ is built entirely out of factorials. Without $n!$ (and the fact $0! = 1$ ) you cannot even write $(0 n)$ or $(n n)$ .

See Factorials for a deeper build.

3. Combinations $(r n)$ — "how many ways to choose $r$ from $n$ "

WHAT. $(r n)$ (also written $^{n} C_{r}$ , read " $n$ choose $r$ ") counts the number of ways to pick $r$ objects out of $n$ when order does not matter.

$(r n) = \frac{n !}{r ! ( n - r )!}$

The picture. In the third figure there are $n = 4$ boxes in a row. You must shade exactly $r = 2$ of them. Count all the distinct shading patterns — that count is $(2 4) = 6$ . Which two boxes, not in what order.

Why the factorial formula? $r!$ in the bottom divides out the orderings of the $r$ chosen boxes (we don't care about order), and $(n - r)!$ divides out orderings of the ones left behind.

Edge cases you'll meet:

WHY the topic needs it. This is the number sitting in front of every term: $(r n)$ counts how many of the $n$ brackets you took $b$ from. It is the heart of the general term. Build it up further in Combinations nCr and see its pattern in Binomial coefficients and Pascal's Triangle.

4. Sigma $\sum$ — the "add-them-all-up" symbol

WHAT. $r = 0 \sum n (expression in r)$ is shorthand for "plug $r = 0$ , then $r = 1$ , ... up to $r = n$ , and add every result."

$\sum_{r = 0}^{3} r^{2} = 0^{2} + 1^{2} + 2^{2} + 3^{2} = 14.$

The picture. $\sum$ is a machine with a counter: it starts the counter at the bottom number, runs the expression, bumps the counter, and stops after the top number — then totals everything.

WHY the topic needs it. The Binomial Theorem is written $(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$ . Reading that line means knowing $\sum$ just gathers one general term for every $r$ from $0$ to $n$ . Each single term inside the sum is the object this whole topic hunts for.

5. Subscripts and the term-number twist: $T_{r + 1}$

WHAT. A subscript is a small label that names which item you mean. $T_{1}, T_{2}, T_{3}, \dots$ are the 1st, 2nd, 3rd terms. The subscript is just an index tag, not a multiplication.

The counting twist. The parent writes the general term as $T_{r + 1}$ , not $T_{r}$ . Here is why, in one picture:

$r$ (b's chosen)	$0$	$1$	$2$	$\dots$
term number $T_{r + 1}$	$T_{1}$	$T_{2}$	$T_{3}$	$\dots$

Because $r$ starts counting at $0$ , the very first term ( $r = 0$ ) is $T_{1}$ . So the term number is always one more than $r$ : term number $= r + 1$ .

WHY the topic needs it. Every "find the $k$ -th term" question is solved by setting $r + 1 = k$ . Get this offset wrong and every answer is one term off.

6. The two-piece bracket $(a + b)^{n}$ — naming the parts

WHAT. $(a + b)^{n}$ is a binomial ("bi" = two) raised to a power. The pieces:

$a$ — the first entry (in Example 1, $a = 2 x$ ),
$b$ — the second entry (in Example 1, $b = 3$ ),
$n$ — the index, how many brackets are multiplied.

The picture. $(a + b)^{n}$ drawn out is $n$ identical copies of the same two-choice bracket standing in a row. Each bracket is a fork: go left to $a$ or right to $b$ .

WHY the topic needs it. Recognising which part is $a$ , which is $b$ , and what $n$ is — including their signs and coefficients — is step 1 of the universal method. Missing the sign in $b = - \frac{1}{x ^{2}}$ is exactly "Slip 3" on the parent page. The full statement lives in Binomial Theorem for positive integer index.

Recall Why do the powers always add to

$n$ ? Each of the $n$ forks contributes exactly one factor. If $r$ of them are $b$ 's and the other $n - r$ are $a$ 's, then $(n - r) + r = n$ : every term carries a total of $n$ factors. That is your instant sanity check on any term.

How these feed the topic

Equipment checklist

Cover the right side and test yourself — you're ready when every line is instant.

What does

x^{p} \cdot x^{q}

equal?

x^{p + q}

— same base, add the powers.

What does

(x^{p})^{q}

equal?

x^{pq}

— power of a power, multiply.

What is

x^{0}

, and why does it matter here?

1

; "term independent of

x

" means the exponent is

0

, i.e.

x^{0} = 1

Rewrite

\frac{1}{x ^{2 r}}

with a negative exponent.

x^{- 2 r}

Compute

5!

5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120

What is

0!

and why?

1

; multiplying no numbers leaves the neutral value

1

Write the formula for

(r n)

\frac{n !}{r ! ( n - r )!}

What does

(r n)

count, and does order matter?

The number of ways to choose

r

items from

n

; order does not matter.

Value of

(0 n)

and

(n n)

Both equal

1

What does

r = 0 \sum n

tell you to do?

Plug in

r = 0, 1, \dots, n

and add all the results.

Which value of

r

gives the 5th term?

r = 4

, because term number

= r + 1

(a + b)^{n}

, what are

a

b

n

called?

First entry, second entry, and index (power).

For

b = - \frac{1}{x ^{2}}

, write

b^{r}

with sign separated.

(- 1)^{r} x^{- 2 r}

Connections

Yeh note Hinglish mein padho →
Binomial Theorem for positive integer index
Binomial coefficients and Pascal's Triangle
Combinations nCr
Factorials
Middle term of a binomial expansion
Greatest coefficient and greatest term