Factorisation (1 Viewer)
- Thread starter Blitz_N7
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I am going to guess that your first line is an intermediate stage in your working for a different question.
If so, could you post the actual question - there might be an easier way to do it.
Ekman
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I got (m+n)(m+n-1)(m+n-2), is there something wrong with the question or am I simply wrong?
Edit: After doing some expansion, I realised that multiplying -n three times will not give positive n^3, so there must be something wrong with the answer
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Much more easily done combinatorially than algebraically.
rand_althor
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It is easier if you consider the coefficient of 
in ^{m+n})
and ^m(1+x)^n)
. You know that the coefficients of in each of these expansions will be equal as ^m(1+x)^n=(1+x)^{m+n})
.
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Blitz_N7
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Thank you! The combinatorial method is to consider picking 3 objects from a total of $m+n$ distinct objects. The number of ways to do this is, on the one hand, $\binom{m+n}{3}$. On the other hand, we can count the number of ways as follows: split the group up in to the first $m$ objects and the last $n$ objects. We can choose 3 by picking $3$ from the first $m$ (done in $\binom{m}{3}$ ways), OR picking 2 from the first $m$ and 1 from the last $n$ (done in $\binom{m}{2}\times \binom{n}{1}$ ways), OR picking 1 from the first $m$ and 2 from the last $n$ (done in $\binom{m}{1}\times \binom{n}{2}$ ways) OR picking $3$ from the last $n$ (done in $\binom{n}{3}$ ways), so the total no. of ways to pick is also $\binom{m}{3} + \binom{m}{2}\times \binom{n}{1}+\binom{m}{1}\times \binom{n}{2} + \binom{n}{3}$. Hence $\binom{m+n}{3}=\binom{m}{3} + \binom{m}{2}\times \binom{n}{1}+\binom{m}{1}\times \binom{n}{2} + \binom{n}{3}$.$)
Blitz_N7
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Thank you!If you still want to know how to do it algebraically:
((m+n)^2 -3(m+n) + 2 )= (m+n)(m+n-1)(m+n-2) )
YouCantMakeMe
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How would you articulate this in an exam? Nice math skills
Probably just write it similarly to that (though you probably wouldn't need full sentences), and maybe draw a diagram if it helps, like:How would you articulate this in an exam? Nice math skills
where each * represents a distinct object.
YouCantMakeMe
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And nice LaTex skills haha.Probably just write it similarly to that (though you probably wouldn't need full sentences), and maybe draw a diagram if it helps, like:
where each * represents a distinct object.