12 x1 t08 02 general binomial expansions (2012)

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<ul><li> 1. General Expansion of Binomials </li> <li> 2. General Expansion of Binomials Ck is the coefficient of x k in 1 x n k </li> <li> 3. General Expansion of Binomials Ck is the coefficient of x k in 1 x n k n n Ck k </li> <li> 4. General Expansion of Binomials Ck is the coefficient of x k in 1 x n k n n Ck k 1 x n nC0 nC1 x nC2 x 2 nCn x n </li> <li> 5. General Expansion of Binomials Ck is the coefficient of x k in 1 x n k n n Ck k 1 x n nC0 nC1 x nC2 x 2 nCn x n which extends to; a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb n </li> <li> 6. General Expansion of Binomials Ck is the coefficient of x k in 1 x n k n n Ck k 1 x n nC0 nC1 x nC2 x 2 nCn x n which extends to; a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb ne.g .2 3 x 4 </li> <li> 7. General Expansion of Binomials Ck is the coefficient of x k in 1 x n k n n Ck k 1 x n nC0 nC1 x nC2 x 2 nCn x n which extends to; a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb ne.g .2 3 x 4C0 2 4 4C1 23 3 x 4C2 2 2 3 x 4C3 23 x 4C4 3 x 4 2 3 4 </li> <li> 8. General Expansion of Binomials Ck is the coefficient of x k in 1 x n k n n Ck k 1 x n nC0 nC1 x nC2 x 2 nCn x n which extends to; a b n nC0 a n nC1a n1b nC2 a n2b 2 nCn1ab n1 nCnb ne.g .2 3 x 4C0 2 4 4C1 23 3 x 4C2 2 2 3 x 4C3 23 x 4C4 3 x 4 2 3 4 16 96 x 216 x 2 216 x 3 81x 4 </li> <li> 9. Pascals Triangle Relationships </li> <li> 10. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 1 </li> <li> 11. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 </li> <li> 12. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 </li> <li> 13. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x k </li> <li> 14. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk </li> <li> 15. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk </li> <li> 16. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk RHS 1 n1Ck 1 1 n1Ck </li> <li> 17. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk RHS 1 n1Ck 1 1 n1Ck n1Ck 1 n1Ck </li> <li> 18. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk RHS 1 n1Ck 1 1 n1Ck n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck </li> <li> 19. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk RHS 1 n1Ck 1 1 n1Ck n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck2 nCk nCnk where 1 k n 1 " Pascal s triangle is symmetrical" </li> <li> 20. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk RHS 1 n1Ck 1 1 n1Ck n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck2 nCk nCnk where 1 k n 1 " Pascal s triangle is symmetrical"3 nC0 nCn 1 </li> <li> 21. Pascals Triangle Relationships1 nCk n1Ck 1 n1Ck where 1 k n 11 x n 1 x 1 x n1 1 x n1C0 n1C1 x n1Ck 1 x k 1 n1Ck x k n1Cn1 x n1 looking at coefficients of x kLHS nCk RHS 1 n1Ck 1 1 n1Ck n1Ck 1 n1Ck n Ck n1Ck 1 n1Ck2 nCk nCnk where 1 k n 1 " Pascal s triangle is symmetrical" Exercise 5B; 2ace, 5, 6ac, 10ac, 11, 143 nC0 nCn 1 </li> </ul>