Third Test Solutions

Solution to Fall 2007 Third Test
1. Test A: 5! - 9x4! + 27x3! - 31x2! + 12.
Test B: 5! - 9x4! + 26x3! - 26x2! + 8.
2. Test A: 26! - 3x23! + 2x20!
Test B: 26! - 3x23! + 2x20! + 21! - 18!
3. Test A: an,k = an-2,k-1 + . . .+ an-5,k-1,
Test B: an,k = an-1,k-1 + . . + an-4,k-1.
4. Test A: a) an = 2an-1 + an-2, b) an = 2an-1 + an-2 - an-3.
Test B: an = an-1 + 2an-3, b) an = an-1 - an-4 + 2an-3.
5. Test A: P(C(7+2-1,2),6) - 7xP(C(6+2-1,2),6) + C(7,2)xP(C(5+2-1,2),6) - C(7,3)xP(C(4+2-1,2),6) + C(&,4)xP(C(3+2-1,2),6).
Test B: Similar to Test A but permutations of length 5, instead of 6.
5; alt Test A: 18!/3!^6 - C(6,1)x6x15!/3!^5 + C(6,2)x6x5x12!/3!^4 . . etc.
Test B: 20!/4!^5 - C(5,1)x5x16!/4!^4 + C(5,2)x5x4x12!/3!^3 . . . etc.

Solution to Fall 2006 Third Test
1. Test A: 5! -8*4! + 21*3! - 20*2! + 6.
Test B: 5! - 8*4! + 22*3! - 24*2! + 9.
2. Test A: 11!/4!4!2! -{2*8!/4!2! + 10!/4!4!} + {2*7!/4! + 5!/2!} - 4!
Test B: 10!/3!2!2! - {2*9!/3!2! + 8!/2!2!} + {2*7!/2! + 8!/3!} - 6!
3. Test A: an,m = an-2,m-1 + an-5,m-1 + an-8,m-1.
Test B: an,m = an-3,m-1 + an-6,m-1 + an-7,m-1.
4. Test A&B: an = 2an-1 + 2an-2; an = 3an-1 - an-3.
5. Test A: C(6*5,15) - 6*(5*4,15).
. Test B: C(5*4,8) - 5*C(4*3,8).

Solution to Fall 2005 Third Test
1. Test A & B: 5! - 8x4! + 21x3! - 20x2! + 6x1!.
2. Test A: 11!/2!2!2! - 3x10!/2!2! + 3x9!/2! - 8!.
Test B: 9!/2!2!2! - 3x8!/2!2! + 3x7!/2! - 6!.
3. Test A: an,k = an-2,k-1 + an-4,k-1 + an-6,k-1.
Test B: an,k = an-3,k-1 + an-5,k-1 + an-7,k-1.
4. Test A: a) an = an-1+ an-2+ an-4, b) an = an-1+ an-2+ an-4 - an-5.
Test B: a) an = an-1+ an-3+ an-5, b) an = an-1+ an-3+ an-5 - an-8.
5. 12!/(2!^6) - C(6,1)10!/(2!^5) + C(6,2)8!/(2!^4) - C(6,3)6!/(2!^3) + C(6,4)4!/2!2! - C(6,5)2!/2! + C(6,6).

Solution to Fall 2004 Third Test
1. 5! - 8x4! + 21x3! - 20x2! + 6x1! .
2. C(37,10)- [C(22,10) + C(25,10) + C(27,10)} + C(15,10) + C(12,10) + C(10,10).
3. an,k = an-2,k-1 + an-3,k-1 + an-4,k-1 + an-5,k-1.
4. a) an = an-1 + an-2 + an-3, b) an = (an-1 - an-6) + an-2 + an-3.
5. 21^n - C(6,1)x20^n + C(6,2)x19^n . . . + C(6,6)x15^n.