33ok i'll post it...
ax1+bx2+cx3+dx4=20
find the non negative integral solns for the the eqn .please explain???
How to find the coefficient of xr term in the expansion of (1+x)-20
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17 Answers
1dude.........this is a basic thing.........its called multinomial theorem ....u must hav learnt it......
33take a small example x1+x2=2
by counting this is
0+2
1+1
2+0
three...
so it is same as coefficient of x2 in the expansion of (1+x+x2)(1+x+x2)..
multiply this simply...(simple multiplication no. (a+b+c)2 formula or binomial..)
and see how coefficients add to give 2..
coeff of x0 frm 1st bracket and x2 from second... is same as 0+2
and coeff of x2 frm 1st bracket and x2 from second... is same as 2+0
this must come as 1.x2+1x2 two times implying 0+2 and 2+0
same for 1+1
there should be one x.x
..
is it now a bit clear...
11i think answer is 23C3
bcos take ax1=X1and so on
the number of solutions will not change
33well...
have read somewhere... but let me think...
when we find for x1+x2+x3=20.. like this... then why we do coeff of x20 in (1+x+x2+...)(...)(...)
same happens here with the only difference that when any x(.) is included in the product.. it takes coefficient int he expression with it..
actually the question is in how many ways sum can be.. 20.. same thing happens here for the powers they add when er multiply... along with the coefficients..
33it is the coefficient of x20 in the expansion of (1+xa+x2a+x3a........)(1+xb+x2b+x3b........)(1+xc+x2c+x3c........)(1+xd+x2d+x3d........)
coefficient of xr in (1+x)-20 20+r-1Cr-1