Find the number of dissimilar terms in the expansion of $(a^{3} + b^{3} + 3 a b (a + b))^{50}$

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Solution

We will first simplify the terms in the bracket so that we can apply binomial expansion.
We know $(a^{3} + b^{3} + 3 a b (a + b))$ = $(a + b)^{3}$ . Therefore, we can
write $(a^{3} + b^{3} + 3 a b (a + b))^{50}$ as $((a + b)^{3})^{50}$ .

It is equal to $(a + b)^{150}$ . There will be a total of (n+1) terms in the expansion of $(a + b)^{n}$ . Therefore the total number terms in the expansion of $(a + b)^{150}$ is 151

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a^{3}

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b^{3}

(a - b)^{3}

+ 3ab(a - b)

Q. Number of dissimilar terms in the expansion of

(x_{1} + x_{2} + . . . . . + x_{n})^{3}

Q. Which of the following equations are correct?

(i) (a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (a b + b c + c a)

(i i) (a + b)^{3} = a^{3} + b^{3} + 3 a b (a - b)

(i i i) (a - b)^{3} = a^{3} - b^{3} - 3 a b (a + b)

(i v) a^{2} - b^{2} = (a + b) (a - b)

Which of the following equation/s is/are correct?

$i) (a + b + c)^{2} = a^{2} + b^{2} + c^{2} - 2 (a b + b c + c a)$
$i i) (a + b)^{3} = a^{3} + b^{3} + 3 a b (a - b)$
$i i i) (a - b)^{3} = a^{3} - b^{3} - 3 a b (a + b)$
$i v) a^{2} - b^{2} = (a + b) (a - b)$

Find the number of dissimilar terms in the expansion of $(a + b)^{100}$

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Find the number of dissimilar terms in the expansion of (a3+b3+3ab(a+b))50 __

Find the number of dissimilar terms in the expansion of $(a^{3} + b^{3} + 3 a b (a + b))^{50}$

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