View source for 2007 UNCO Math Contest II Problems/Problem 5

== Problem ==


Ten different playing cards have the numbers
<math>1, 1, 2, 2, 3, 3, 4, 4, 5, 5</math> written on them
as shown. Three cards are selected at random
without replacement. What is the
probability that the sum of the
numbers on the three cards is divisible by <math>7</math>?

<asy>
size(200,100);
path Card=(arc((.2,.2),.2,180,270)--(1.8,0)--arc((1.8,.2),.2,270,360)--(2,2.8)--arc((1.8,2.8),.2,0,90)--(.2,3)--arc((.2,2.8),.2,90,180)--(0,.2));
draw(Card,black);
draw(shift(3,0)*Card,black);
draw(shift(6,0)*Card,black);
draw(shift(9,0)*Card,black);
draw(shift(12,0)*Card,black);
draw(shift(0,-4)*Card,black);
draw(shift(3,-4)*Card,black);
draw(shift(6,-4)*Card,black);
draw(shift(9,-4)*Card,black);
draw(shift(12,-4)*Card,black);
MP("1",(.25,2.1),N);MP("1",(1.75,-.1),N);
MP("1",(.25,2.1-4),N);MP("1",(1.75,-4.1),N);
MP("\spadesuit",(1,1),N);MP("\clubsuit",(1,1-4),N);

MP("2",(3.25,2.1),N);MP("2",(4.75,-.1),N);
MP("2",(3.25,2.1-4),N);MP("2",(4.75,-4.1),N);
MP("\spadesuit",(4,1.75),N);MP("\clubsuit",(4,1.75-4),N);
MP(180,"\spadesuit",(4,.25),N);MP(180,"\clubsuit",(4,.25-4),N);


MP("3",(6.25,2.1),N);MP("3",(7.75,-.1),N);
MP("3",(6.25,2.1-4),N);MP("3",(7.75,-4.1),N);
MP("\spadesuit",(7,1.75),N);MP("\clubsuit",(7,1.85-4),N);
MP("\spadesuit",(7,1),N);MP("\clubsuit",(7,1-4),N);
MP(180,"\spadesuit",(7,.25),N);MP(180,"\clubsuit",(7,.25-4),N);

MP("4",(9.25,2.1),N);MP("4",(10.75,-.1),N);
MP("4",(9.25,2.1-4),N);MP("4",(10.75,-4.1),N);
MP("\spadesuit",(10-.3,1.75),N);MP("\clubsuit",(10-.3,1.75-4),N);
MP(180,"\spadesuit",(10-.3,.25),N);MP(180,"\clubsuit",(10-.3,.25-4),N);
MP("\spadesuit",(10+.3,1.75),N);MP("\clubsuit",(10+.3,1.75-4),N);
MP(180,"\spadesuit",(10+.3,.25),N);MP(180,"\clubsuit",(10+.3,.25-4),N);

MP("5",(12.25,2.1),N);MP("5",(13.75,-.1),N);
MP("5",(12.25,2.1-4),N);MP("5",(13.75,-4.1),N);
MP("\spadesuit",(13-.3,1.75),N);MP("\clubsuit",(13-.3,1.75-4),N);
MP("\spadesuit",(13,1),N);MP("\clubsuit",(13,1-4),N);
MP(180,"\spadesuit",(13-.3,.25),N);MP(180,"\clubsuit",(13-.3,.25-4),N);
MP("\spadesuit",(13+.3,1.75),N);MP("\clubsuit",(13+.3,1.75-4),N);
MP(180,"\spadesuit",(13+.3,.25),N);MP(180,"\clubsuit",(13+.3,.25-4),N);

</asy>



== Solution ==
<math>\frac{2}{15}</math>

There are five ways to achieve a sum divisible by 7; 115 (2 ways), 133 ( 2 ways), 124 (8 ways), 223 (2 ways), 455 (2 ways). Hence, there are 16 favorable ways out of <math>120=\binom{10}{3}</math>.

== See Also ==
{{UNCO Math Contest box|n=II|year=2007|num-b=4|num-a=6}}

[[Category:Introductory Probability Problems]]