A handful of questions

1) 23,760
2) 81
3) 11,700
4) 105

Well, that clears it up then.

1) a)Novel and dictionary selection
use combinations--n!/r!/(n-r)!

Novels n=11 & r=4 (11!/4!/7!)=11*5*3*2=330
Dictionary n=3, r=1 (3!/1!/2!)=3

Use permutations (n!/r!)

b) Arrangement on shelf--the dictionary is fixed so n=1 permutations possible 1 & 1!/1!=1
The novels aren't so fixed so n=4 and 1 possible so (4!/1!)=4!=24

c) The total number of possibilities is the product of the possible selections and arrangement

P=330*3*1*24=23760

2) (2002)^2=(2*7*11*13)^2 note all factors are prime
Since there are 4 distinct primes in 2002. The square has four duplicated primes
Using combinations (again) if there were strictly 8 distinct there would be
C=8!/1!/7!+8!/2!/6!+8!/3!/5!+8!/4!/4!+8!/5!/3!+8!/6!/2!+8!/7!/1!=324

The problem is that when 2002 is squared this repeated primes. The number is reduced by 2^2=4
And 324/4=81

Note if you were to include factors 1, 2002, and 4008004 there would be three more and the total number is 81+3=84

3) n=11 p=-210 & n=30, p=-153
set this up as p=x+rn then -210=x+11r & -153=x+30r

setting x on one side and equating t0o linear terms then 11r+210=30r+153 or 19r=57 & r=3. Using this r then x is -243
The series is then p=-243+3n
Summing this series over the first 200 terms is then
Sum=Sum(-243+3n)=200(-243)+3Sum (r )=-48600+3Sum(r )
Now Sum(r ) over 200 digits is (200)(200+1)/2=20100
So the sum over 200 digits is -48600+3*20100=11700

4) Classic parallel problem. Set it up as 1/A+1/B=1/7 and B=2A so 1/A+1/2A=1/7
or 2/2A+1/A=3/2A=1/7 rearranging 2A=21 and A=10.5. So A can dig 10 holes in 105 hrs

Rap