MathDB

10

Part of 2007 AIME Problems

Problems(2)

Grid Combinatorics

Source: AIME I 2007 #10

3/15/2007
In the 6×4 6\times4 grid shown, 12 12 of the 24 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let N N be the number of shadings with this property. Find the remainder when N N is divided by 1000 1000. [asy]size(100); defaultpen(linewidth(0.7)); int i; for(i=0; i<5; ++i) { draw((i,0)--(i,6)); } for(i=0; i<7; ++i) { draw((0,i)--(4,i)); }[/asy]
AoPSwikiAMCAIMEAIME I
Probability of sets

Source: AIME II 2007 #10

3/29/2007
Let SS be a set with six elements. Let PP be the set of all subsets of S.S. Subsets AA and BB of SS, not necessarily distinct, are chosen independently and at random from PP. the probability that BB is contained in at least one of AA or S−AS-A is mnr,\frac{m}{n^{r}}, where mm, nn, and rr are positive integers, nn is prime, and mm and nn are relatively prime. Find m+n+r.m+n+r. (The set S−AS-A is the set of all elements of SS which are not in A.A.)
AIMEcombinatorics