The number of combinations?

The number of combinations?

Every box has the same number of balls, balls in each box are distinct.
Box 1 contain balls: a1, b1, c1, and d1 Box 2 contain balls: a2, b2, c2,
and d2 The total number of balls will be in this example n=No. of
boxes×No. of balls in each box=2×4=8. We can choose any number from the
available balls, but in two conditions:
1- the number must be even
2- every 2 balls must be from the same box
Example: We want to know the number of possible combinations of choosing
two balls (k=2) for the above example: From the above conditions, the
available combinations are:
(a1,b1 ),(a1,c1 ),(a1,d1 ), (b1,c1 ),(b1,d1 ), (c1,d1) (a2,b2 ),(a2,c2
),(a2,d2 ), (b2,c2 ),(b2,d2 ), (c2,d2) The number of combinations is 12.
If we can choose the two balls one from each box or the two from the same
box, then the number of combinations will be (8 choose 2)=28 (12 in our
case)
also, if i want to choose 4 balls (two and two) We can choose the four
balls from box1 or choose the 4 balls in the other box or choose two balls
from box1 and the other two balls from box2, so the possible combinations
are:
(a1 b1,c1 d1 ),(a1 b1,a2 b2 ),(a1 b1,a2 c2 ),(a1 b1,a2 d2 ),(a1 b1,b2 c2
),(a1 b1,b2 d2 ) (a1 b1,c2 d2 ),(a1 c1,a2 b2 )…..(a1 c1,c2 d2 ),(a1 d1,a2
b2 )…(a1 d1,c2 d2 ) ,(b1 c1,a2 b2 )…(b1 c1,c2 d2 ),(b1 d1,a2 b2 )…(b1
d1,c2 d2 ),(c1 d1,a2 b2 )… (c1 d1,c2 d2 ),(a2 b2,c2 d2) The number of
combinations is 38.
if we choose it one by one the number of combinations will (8 choose 4)=70
(38 in our case)