SOLUTIONS TO PREVIOUS SECOND TESTS

Solution to Fall 2010 Test 2
1.(8 pt) Test A C(16+7-1,16) - 7xC(7+7-1,7), Test B C(15+6-1,15) - 6xC(7+6-1,7).
2. (8 pt) Test A a) e1+ e2 + . . + e6 = 27, ei>=4; b) (x^4+ x^5 . .)^6, coef of x^27, c) C(3+6-1,3), Test B a) e1+ e2+ . . e4 =29, ei>=5, b) (x^5 + x^6 + ..)^4, coef of x^29, c) C(9+4-1,9).
3. (7 pt) 21!5!.
4. (8 pt) Test A C(13,3)C(4,2)^3*C(10,2)C(4,1)^2, Test B: C(13,3)C(4,3)^3*C(10,2)C(4,1)^2.
5. (10 pt) Test A 9!/2!2!2! + 9!/2!2! - 8!/2!, Test B: 8!/2!2! + 8!/2! - 7!.
NOTE: This answer is actually not fully correct-- one has to worry about having two LPs. The right answer for arrangements with at least one LP (and possibly also MP) is 9!/2!2! - 8!/2!2!.
6. (14 pt) Test A C((8-3)+4-1,(8-3))*1*4!/2!, Test B C((8-4)+5-1,(8-4))*1*5!/3!.
7. (11 pt) Test A sum(k=0 to 3){C(8,k)*2^k*4^(8-k)}, Test B sum(k=0 to 3){C(9,k)*2^k*3^(9-k)} .

Solution to Fall 2009 Test 2
1.(8 pt) C(9+5-1,9) - 5xC(5+5-1,5) + C(5,2)xC(1+5-1,1).
2. (8 pt) a) e1+ e2 + . . + e8 = 25, ei>=2; b) (x^2+ x^3 + x^4 . .)^8, coef of x^25, c) C(9+8-1,9).
3. (7 pt) C(13,4)*4^4/C(52,4).
4. (8 pt) C(6,r)*9^(6-r).
5. (10 pt) 11!/(3!2!) - 10!/(2!2!).
6. (14 pt) 2C(3+3-1,3)C(8,6)*1*2!.
7. (10 pt) P(C(6,2),5).

Solution to Fall 2008 Test 2
1.(8 pt) C(13+8-1,13) - 8xC(8+8-1,8) + C(8,2)xC(3+8-1,3).
2. (8 pt) Test A: a) e1+ e2 + . . + e6 = 23, ei>=3; b) (x^3+ x^4 + x^5 . .)^6, coef of x^23, c) C(5+6-1,5).
3. (7 pt) Test A: 10^4.
4. (8 pt) Test A: C(13,1)xC(4,3)xC(12,5)xC(4,1)^5/C(52,8).
5. (10 pt) Test A: 2(n-4)(n-2)!
6. (15 pt) Test A: C(7+2-1,8)xC(8,4)x[4!/2!2!-1]x4!
7. (10 pt) Test A and Text B: C(4,1)^4xC(36,2) + 4[C(4,2)xC(4,1)^3xC(36,1)] + C(4,2)[C(4,2)^2 x C(4,1)^2]+ 4{C(4,3)C(4,1)^3].