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Try this beautiful problem based on expansion, useful for ISI B.Stat Entrance

## Expansion Problem | ISI B.Stat TOMATO 102

The number of terms in the expansion of $[(a+3b)^2 (a-3b)^2]^2$ , when simplified, is

• 4
• 5
• 6
• 7

### Key Concepts

Algebra

multiplication

Expansion

But try the problem first…

Answer: $5$

Source

TOMATO, Problem 102

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

The given expression is $[(a+3b)^2 (a-3b)^2]^2$.we have to find out number of terms in the expansion of $[(a+3b)^2 (a-3b)^2]^2$.

Now , $[(a+3b)^2 (a-3b)^2]^2$ $\Rightarrow [(a^2-9b^2)^2]^2$ $\Rightarrow (a^2-9b^2)^4$

Can you now finish the problem ……….

Second Hint

Now in the equation $(a+b)^2=a^2+2ab+b^2$ i.e total numbers of terms are $3$

$(a+b)^3=a^3+3a^2b+3ab^2+b^3$ i.e total numbers of terms are $4$

………….

………….

$(a + b)^n = a^n + { n \choose 1} a^{n-1}b + { n \choose 2 } a^{n-2}b^2 + … + { n \choose {n-1}}ab^{n-1} + b^n$ i.e total numbers of terms are $n+1$

Similarly In this expression $(a^2-9b^2)^4$ ,The power is $4$.Therefore we say that after the expansion total number be $5$.