Homework Assignment Eleven Quick Answers



1. 8.2.2 6**10-C(6,2)5**10+C(6,3)4**10-..

2. 8.2.4 prime factors: 2,3,5,7, so: 420  (240+140+82+60)+(divisors 
(multiples) of pairs)  (multiples of triples)+(multiple of 
quadruples=210)

3. 8.2.12 
  10**4  C(10,1)*9**2*4!/2/!+C(10,2)4!/2!2!

4. 8.2.18  Assume n is even; n odd is similar:
  n!  C(n/2, 1)(n-1)! + (n/2 2)(n-2)!..C(n/2,n/2)(n-n/2)!

5. 8.2.28
  (2n)!-C(n,1)*2*(2n-1)!+C(n,2)*2**2*(2n-2)!..
      +(-1)**k*C(n,k)*2**k*(2n-k)!

6. 8.3.2-a,d for a: permute columns so that there are three disjoint 
clumps. Then: (1+x)(1+2x)(1+4x+3x**2)

7. 8.3.4 Rework the board to obtain two distinct disjoint patterns, 
each of which have polynomial (1+4x+2x**2). Square this to get: 
(1+8x+20x**2+16x**3+4x**4). Use this to count:

5!-8*4!+20*3!-16*2!+3*4!


8. 8.3.12 As done in class: For a) move down the top row, doing the standard expansion of
each cell. When you finish with the row, you'll have the RHS of the equality. 
For b) on the LHS, the coefficient of x**(k-1) will be k*Ri,j for k rooks. This is like
placing k rooks, and making one distinguished (which can happen in k ways). The RHS says:
make  one of m*n choices for the distinguished rook, then place the others.

-- RobbieMoll - 12 Dec 2006

Topic revision: r2 - 2006-12-12 - RobbieMoll
 
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