Question 1194941
.
In choosing what music to play at a charity fund raising event, 
Cory needs to have an equal number of symphonies from Haydn, Mendelssohn, and Mahler. 
If he is setting up a schedule of the 12 symphonies to be played, 
and he has 104 Haydn, 17 Mendelssohn, and 9 Mahler symphonies from which to choose, 
how many different schedules are possible? 
Express your answer in scientific notation rounding to the hundredths place.
~~~~~~~~~~~~~~~~


<pre>
Cory can choose 4 symphonies from 104 symphonies by Haydn in  {{{C[104]^4}}} = 4598126 different ways.

Cory can choose 4 symphonies from 17 symphonies by Mendelssohn in  {{{C[17]^4}}} = 2380 different ways.

Cory can choose 4 symphonies from 9 symphonies by Mahler in  {{{C[9]^4}}} = 126 different ways.


So, Cory can combine  {{{C[104]^4 * C[17]^4 * C[9]^4}}} = 4598126 * 2380 * 126 = 1378886024880

different sets of 12 symphonies taking 4 symphonies of each of the three famous composers.


    +-----------------------------------------------------------------------+
    |    But this number is not yet the number of all possible schedules.   |
    +-----------------------------------------------------------------------+


To get the number of all possible schedules, we should take every of these 1378886024880 sets

and make it a subject of all possible  12! = 12*11*10*9*8*7*6*5*4*3*2*1 = 479001600  permutations.



After this permutization of each of 1378886024880 sets, we get the total number of all possible schedules

    {{{C[104]^4 * C[17]^4 * C[9]^4}}} * 12! = 1378886024880 * 479001600 = {{{6.60489*10^20}}} schedules.    


You can round this number to the requested form and get the  <U>ANSWER</U> :   there are  {{{6.60*10^20}}}  different possible schedules.
</pre>

Solved.