7.1 Q-12 | Mathematical Induction And Binomial Theorem | Mathematics 11

When $b$ is negative, alternate signs appear: $+, -, +, -, \dots$
Always simplify numerical coefficients: $(r n) \times (coefficient of a)^{n - r} \times (coefficient of b)^{r}$ .
The expansion of $(a + b)^{n}$ has exactly $n + 1$ terms.

Exercise 7.1 — Question 12

Relevant SLOs: M-11-A-34, M-11-A-37

For any positive integer $n$ :

$(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$

where the general term (the $(r + 1)$ th term) is:

$T_{r + 1} = (r n) a^{n - r} b^{r}$

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

Expand and simplify the given binomial expression using the Binomial Theorem.

Note: The specific expression for Q-12 should be taken from your FBISE textbook (Chapter 7, Exercise 7.1, Q.12).

Step 1: Identify $a$ , $b$ , and $n$ from the expression $(a + b)^{n}$ .

Step 2: Write out all terms using: $T_{r + 1} = (r n) a^{n - r} b^{r}, r = 0, 1, 2, \dots, n$

Step 3: Evaluate each binomial coefficient $(r n)$ .

Step 4: Simplify each term by combining powers and constants.

Step 5: Collect and write the final expanded form.

Expand $(2 x - y)^{4}$ .

Here $a = 2 x$ , $b = - y$ , $n = 4$ .

$(2 x - y)^{4} = \sum_{r = 0}^{4} (r 4) (2 x)^{4 - r} (- y)^{r}$

$r$	$(r 4)$	$(2 x)^{4 - r}$	$(- y)^{r}$	Term
0	1	$16 x^{4}$	$1$	$16 x^{4}$
1	4	$8 x^{3}$	$- y$	$- 32 x^{3} y$
2	6	$4 x^{2}$	$y^{2}$	$24 x^{2} y^{2}$
3	4	$2 x$	$- y^{3}$	$- 8 x y^{3}$
4	1	$1$	$y^{4}$	$y^{4}$

$(2 x - y)^{4} = 16 x^{4} - 32 x^{3} y + 24 x^{2} y^{2} - 8 x y^{3} + y^{4}$

When $b$ is negative, alternate signs appear: $+, -, +, -, \dots$
Always simplify numerical coefficients: $(r n) \times (coefficient of a)^{n - r} \times (coefficient of b)^{r}$ .
The expansion of $(a + b)^{n}$ has exactly $n + 1$ terms.