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For general n, a permutation i, j, h, k, ... , r, s of 1, 2, 3, ... , n is associated with the product
 
p = [(j - i)][(h - i)(h - j)][(k - i)(k - j)(k - h)] ...[(s - i)(s - j)(s - h)(s - k)...(s - r)]
 
of all the differences of two of i, j, h, k, ... , r, s in which the number that appears first is subtracted from the other. If the permutation i, j, h, k, ... , r, s is written in the notation a1, a2, a3, a4, ... , an - 1, an, then the product p takes the form


 
If the product p is positive, the permutation is even; if p is negative, the permutation is odd.
 

Problems for Chapter 7
 
1. Write out all the combinations of two letters chosen from a, b, c, d, and e.
 
2. Write out all the combinations of three letters chosen from a, b, c, d, and e.
 
3. Write out all the permutations of two letters chosen from a, b, c, d, and e.
 
4. Write out all the permutations of three letters chosen from a, b, c, d, and e.
 
5. Find the positive integer that is the coefficient of x3y7z2 in (x + y + z)12.

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Tuesday, May 12, 1998