Question 943372
College randomly assigns students to 5 dorms. if 3 friends apply to live in these dorms, what is probability they are assigned to same dorm? What is probability they are assigned to different dorms? 
<pre>
Suppose the friends are A, B and C.

{{{P(matrix(1,5,  

(matrix(7,1,A, will, be, assigned, to, some, dorm))^"",

AND,

(matrix(8,1,B, will, be, assigned, to, that, same, dorm))^"",

AND,

(matrix(8,1,C, will, be, assigned, to, that, some, dorm))^""

))}}} {{{""=""}}}

{{{P(matrix(7,1,A, will, be, assigned, to, some, dorm))*

P(matrix(8,1,B, will, be, assigned, to, that, same, dorm))*

P(matrix(8,1,C, will, be, assigned, to, that, same, dorm)) }}} {{{""=""}}}

{{{(1)*(1/5)*(1/5)}}} {{{""=""}}} {{{1/25}}}

[Notice that the first probability is 1 because it is certain
that A will be assigned to some dorm.]

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</pre>What is probability they are assigned to different dorms?<pre> 
{{{P(matrix(1,5,  

(matrix(7,1,A, will, be, assigned, to, some, dorm))^"",

AND,

(matrix(10,1,B, will, be, assigned, to, a, different, dorm, from, A))^"",

AND,

(matrix(12,1,C, will, be, assigned, to, a, different, dorm, from,A,or,B))^""

))}}} {{{""=""}}}

{{{P(matrix(7,1,A, will, be, assigned, to, some, dorm))*

P(matrix(10,1,B, will, be, assigned, to, a, different, dorm, from, A))*

P(matrix(12,1,C, will, be, assigned, to, a, different, dorm, from, A,or,B)) }}} {{{""=""}}}

{{{(1)*(4/5)*(3/5)}}} {{{""=""}}} {{{12/25}}}

Edwin</pre>