SOLUTIONS TO PREVIOUS TESTS

Solution to Fall 2009 Test 2
1.(8 pt) Test A: C(9+5-1,9) - 5xC(5+5-1,5) + C(5,2)xC(1+5-1,1); Test B: C(10+7-1,10) - 7xC(6+7-1,6) + C(7,2)xC(2+7-1,2) .
2. (8 pt) Test A: a) e1+ e2 + . . + e8 = 25, ei>=2; b) (x^2+ x^3 + x^4 . .)^8, c oef of x^25, c) C(9+8-1,9); Test B: a) e1+ e2 + . . + e4 = 27, ei>=4; b) (x^4+ x^5 + x^6 . .)^4, c oef of x^25, c) C(11+4-1,11).
3. (7/8 pt) Test A: C(13,4)*4^4/C(52,4); Test B: C(9,k)*9^(9-k).
4. (8/7 pt) Test A: C(6,5)*9^(6-r); Test B: C(13,3)*4^3/C(52,3).
5. (10 pt) Test A: 11!/(3!2!) - 10!/(2!2!); Test B: 8!/(2!2!) - 7!/2!
6. (14 pt) Test A: 2C(3+3-1,3)C(8,6)*1*2!; Test B:2C(6+3-1,6)C(11,3)3!/(2!1!)
7. (10 pt) Test A: P(C(6,2),5); Test B: P(C(5,2),7).

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].


Solution to Fall 2007 Test 2
1. (8pt) C(5+13-1,13) - 5*C(5+7-1,7) + C(5,2)*C(5+1-1,1)
2. (8pt) a) e1 + e2 + . . +e6 = 31, ei>=2, b) (x^2 + x^3 + . . )^6, coef.of x^31, c) C(6+19-1,19);
3. (8pt) C(13,1)C(4,4)C(12,2)C(4,1)^2/C(52,6)
4. (8pt) C(5,2)C(3,1)24^2
5. (8pt) C(3,2)*7!/2!.
6. (14pt) C(5+(6-4)-1,(6-4))*4!/2!*(6!/2!-1)
7. (12pt) First pick subset of positions where A's and B's will go; put an A in the first of these positions and then distribute the remaining A's and B's in >he other positions; Test A: C(16,11)*10!/4!6!; Test B: C(17,12)*11!/3!8!.

Solution to Fall 2006 301 Test 2
1. C(7+18-1,18) - 7C(7+9-1,9) + C(7,2)*1.
2. e1 + e2 + . . e6 = 29, 4 =< ei; (x^4 + x^5 + . . .)^6, coef x^29; C(6+5-1,5)
3. 5!*21!
4. C(13,3)C(4,2)^3*40
5. 3(8!/3!3!2! + 8!/4!2!2!)
6. C(4+(8-3)-1,(8-3))*1*4!/2!
7. 4*C(4+(11-5)-1,(11-5))-C(4,2)C(4+(11-10)-1,(11-10)).