MATH:4050: Introduction to Discrete Mathematics. Homework 6 due 10/06/17. 1. Find the coecient of x41 x22 x3 in the expansion of (2x1 6x2 8x3 )7 . 2. Use Newton's binomial theorem to compute 30 to 4 decimal places. 3. Use Newton's binomial theorem to compute 3 30 to 4 decimal places. 4. Use the binomial theorem to prove n ( ) n k 2 = 3n . k k=0 Generalize to evaluate n ( ) n k r k k=0 for any real number r. Justify your result. 5. Evaluate the sum ( ) n (1) 10k . k k=0 n k 6. Express the sum ( ) ( ) ( ) ( ) n n n n +3 +3 + k k1 k2 k3 as a single binomial coecient. 7. A mouse eats a cheese cube consists of 4 4 4 smaller cubes of cheeses. A mouse starts at the corner cheese cube and then eat an adjacent cheese cube (no diagonal eating allowed). It will eat the opposite corner cheese cube last. In how many ways the mouse can eat this cheese? 8. Prove the following identity ( )( ) ( )( ) ( )( ) ( )( ) ( ) n m n m n m n m n+m + + + + = 0 k 1 k1 2 k2 k 0 k by using (a) algebra; (b) combinatorial reasoning(committee forming arguments); 9. Prove the following identity ( ) ( ) ( ) ( ) ( ) n n3 n1 n2 n3 = + + k k k1 k1 k1 by using (a) algebra; (b) combinatorial reasoning(committee forming arguments); 10. Prove the following identity by integrating the binomial expansion: ( ) ( ) ( ) 1 n n 1 n 1 2n+1 1 1+ + + + = , 2 1 3 2 n+1 n n+1 for every positive integer n. 11. Evaluate the sum for every positive integer n ( ) ( ) ( ) ( ) 1 n 1 n 1 n 1 n n 1 + + + (1) . 2 1 3 2 4 3 n+1 n 12. Prove that for every positive integer n, ( ) ( ) ( ) ( ) n n n n n1 2 +3 + (1) n = 0. 1 2 3 n