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Let a,b,c,d in N and a lt b lt c lt d ...

Let a,b,c,d `in ` N and `a lt b lt c lt d ` such that the equation `|x-a|+|x-b|+|x-c|+|x-d|=20-2(a+b)` has infinite solutions then find the number of possible quadruplets (a,b,c,d).

Text Solution

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552
f(b)=f( c)=d+c-a-b=20-2(a+b)
`:. a+b+c+d=20`
Let a=x,b=a+y,c=b+z, d=c+w
`:.` 4x+3y+2z+w=20
As `23xx4!=552`.
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