By Abraham P Hillman

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N & a & b)! a $ 0, b $ 0, and n $ a % b. 14. Given that n = a + b + c + d and that a, b, c, and d are non-negative integers, show that n a n&a b n&a&b c n&a&b&c d 15. Express j [a % (k & 1)d] as a polynomial in n. n k'1 n 16. Express A (2k) compactly without using the A notation. k'1 n n&1 k'1 j'0 17. Show that A a k ' A aj%1. 18. Show that j bk ' j bi&2. n&2 n k'1 i'3 48 ' n! d! 19. Evaluate j ai j bi 2 2 i'1 i'1 n A ai 20. Show that i'1 n A bi i'1 and j (aib i ) and show that they are not always equal.

30. Find x, given that 1/30, 1/x, and 1/20 are in arithmetic progression. What is the relation between x and the answer to Part (b) of problem 29? 31. Verify the factorization 1 - x7 = (1 - x)(1 + x + x2 + x3 + x4 + x5 + x6) and use it with x = 1/2 to find a compact expression for 1% 1 1 % 2 2 2 % 1 2 3 % 1 2 4 % 1 2 5 % 1 2 6 . 32. Use the factorization 1 + x99 = (1 + x)(1 - x + x2 - x3 + x4 - ... + x98) to find compact expressions for the following sums: (a) 1 - 5-1 + 5-2 - 5-3 + ... - 5-97 + 5-98.

N % m) . m % 1 Prove it for general m. * 33. Prove that n5 - n is an integral multiple of 30 for all integers n. * 34. Prove that n7 - n is an integral multiple of 42 for all integers n. * 35. Show that every integer from 1 to 2n+1 - 1 is expressible uniquely as a sum of distinct powers of 2 chosen from 1, 2, 22, ... , 2n. 36. Show that every integer s from & * 3n%1 & 1 3n%1 & 1 to has a unique expression of the 2 2 form s ' c0 % 3c1 % 32c2 % ... , cn is 0, 1, or -1. 39 Chapter 6 THE BINOMIAL THEOREM n r In Chapter 1 we defined as the coefficient of an-rbr in the expansion of (a + b)n, and tabulated these coefficients in the arrangement of the Pascal Triangle: n Coefficients of (a + b)n 0 1 1 1 2 1 3 1 4 1 5 1 6 ...

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