Some Important Points
Category : JEE Main & Advanced
(1) Pascal's Triangle
| 1 | \[{{(x+y)}^{0}}\] | |||||||||||
| 1 | 1 | \[{{(x+y)}^{1}}\] | ||||||||||
| 1 | 2 | 1 | \[{{(x+y)}^{2}}\] | |||||||||
| 1 | 3 | 3 | 1 | \[{{(x+y)}^{3}}\] | ||||||||
| 1 | 4 | 6 | 4 | 1 | \[{{(x+y)}^{4}}\] | |||||||
| 1 | 5 | 10 | 10 | 5 | 1 | \[{{(x+y)}^{5}}\] |
Pascal's triangle gives the direct binomial coefficients.
Example : \[{{(x+y)}^{4}}={{x}^{4}}+4{{x}^{3}}y+6{{x}^{2}}{{y}^{2}}+4x{{y}^{3}}+{{y}^{4}}\].
(2) Method for finding terms free from radicals or rational terms in the expansion of \[{{({{a}^{1/p}}+{{b}^{1/q}})}^{N}}a,\,b\,\in \]prime numbers: Find the general term \[{{T}_{r+1}}{{=}^{N}}{{C}_{r}}{{({{a}^{1/p}})}^{N-r}}{{({{b}^{1/q}})}^{r}}{{=}^{N}}{{C}_{r}}\,{{a}^{\frac{N-r}{p}}}.{{b}^{\frac{r}{q}}}\]
Putting the values of \[0\le r\le N\], when indices of \[a\] and \[b\] are integers.
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