document.write( "Question 628135: I have some equations in the following format (finding the nth term of an algebraic expression)\r
\n" ); document.write( "\n" ); document.write( "( n ) a^(n-(r-1)) * b^(r-1)
\n" ); document.write( " r-1\r
\n" ); document.write( "\n" ); document.write( "I don't understand what the (n & r-1) terms in the parenthesis at the beginning of the expression are. From the solution, it suggests that the first term ends up resolving to a division of factorials such as:\r
\n" ); document.write( "\n" ); document.write( "(6! / ((6-3)! * 3!)) where n=6 and r=4...\r
\n" ); document.write( "\n" ); document.write( "I just dont understand how they got from the first form to the factorial form. \n" ); document.write( "

Algebra.Com's Answer #395467 by solver91311(24713) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
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\n" ); document.write( "\n" ); document.write( "I think what you are looking at is

(read: \"n choose r - 1\") which is the number of combinations of

things taken

at a time. Such as, how many ways can I choose from 10 different books on a shelf if I take them 3 at a time and I don't care what order the three are in when I get them in my hand?\r
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\n" ); document.write( "\n" ); document.write( "In general,

is calculated by

. Hence, your

is exactly correct given

and

.\r
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\n" ); document.write( "\n" ); document.write( "Here's the logic: Let's say you have 6 things and want to choose 3 of them. There are 6 ways to choose the first one, then since you didn't replace the first one you chose, there are 5 ways to choose the second one for each one of the 6 ways to choose the first one. Then there are 4 ways to choose the third one for each of the 30 ways to pick the first two, which works out to

which comes from

. But that number is too large by a factor of the number of ways to arrange the three things in your hand, namely

, and that is where the other denominator factor comes from.\r
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\n" ); document.write( "\n" ); document.write( "You might want to compare this to the number of permutations of

things taken

at a time. With permutations, order matters. Such as you have 20 people in your club and you want to know how many different ways you can select a President, Secretary, and Treasurer. Here order matters because Suzy being the president is a different outcome than Suzy being the Secretary, for example. Permutations are calculated

. See the difference?\r
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\n" ); document.write( "\n" ); document.write( "By the way, this is also the

th coefficient (counting from zero) of the binomial expansion of

and the

th element of the

th row of Pascal's Triangle.\r
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\n" ); document.write( "\n" ); document.write( "John
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\n" ); document.write( "My calculator said it, I believe it, that settles it
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$\"The$

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