1.
(a) No repeated digits. P(9,3)/9^3=9*8*7/9^3=56/81
(b) All three digits the same. C(9,1)/9^3=9/9^3=1/81
(c) Exactly one repeated digit. 1- 56/81-1/81=24/81=8/27
2.
(a) At least 2 yellow marbles.
1-C(6,5)/C(10,5)-C(4,1)*C(6,4)/C(10,5)=1-1/42-10/42=31/42
(b) At least 2 yellow marbles and at least 1 green marble.
1-C(7,5)/C(10,5)-C(6,5)/C(10,5)-C(4,1)*C(6,4)/C(10,5)
=1-7/84-1/42-10/42
=55/84
(c) All four colors represented.
C(1,1)*C(2,1)*C(3,1)*C(4,1)/C(10,5)
=1*2*3*4*(1*2*3*4*5)/(10*9*8*7*6)
=2/21
(d) At least two colors represented.
1-C(9,5)/C(10,5)-C(8,5)/C(10,5)-C(7,5)/C(10,5)-C(6,5)/C(10,5)
=1-1/2-2/9-1/12-1/42
=209/252
(e) At most three colors represented.
1-C(1,1)*C(2,1)*C(3,1)*C(4,1)/C(10,5)
=1-2/21
=19/21