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lecture_04
Kurt Robert Rudolph edited this page Jun 13, 2012
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16 revisions
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recall
- [ (x + y)^n = \sum\limit_{k=0}^n { \frac{ n!}{ k!(n - k)!} x^k y^{n-k} ]
- [ = \sum\limits_{ a + b - n , a \ge 0, b \ge 0}{ \frac{ n!}{ a! b!} x^a y^b ]
- therefore
- [ (x_1 + x_2)^n = \sum\limits_{ n_1 + n_2 = n,n_1 \ge 0, n_2 \ge 0}{ \frac{ n!}{ n_1! n_2!} x_1^{n_1} x_2^{n_2} ]
- [ (x + y)^n = \sum\limit_{k=0}^n { \frac{ n!}{ k!(n - k)!} x^k y^{n-k} ]
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Trinomial Theorem
- [ (x_1 + x_2 + x_3)^n = \sum\limits_{ n_1 + n_2 + n_3 = n, n_i \ge 0}{ \frac{ n!}{ n_1! n_2! n_3!} x_1^{n_1} x_2^{n_2} x_3^{n_3} } ]
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Multinomial Theorem
- [ \left( \sum\limits_{ i = 1}^I{ x_i} \right)^n = \sum\limits_{ n_1 + \cdots n_I = n, n_i \ge 0}{ \frac{ n!}{ n_1! n_2! \cdots n_I!} x_1^{n_1} \cdots x_I^{n_I} ]