Question 1044457
nCr:nC(r-1) = 2:3 , nC(r-2):nC(r-1) = 4:3 find n and r
<pre><b>
Simplify the first equation:

{{{(nCr)/(nC(r-1))}}}{{{""=""}}}{{{2/3}}}

{{{3(nCr^"")}}}{{{""=""}}}{{{2(nC(r-1)^"")}}}

{{{(3n!)/(r!(n-r)!^""))}}}{{{""=""}}}{{{2n!/((r-1)!(n-(r-1)^"")!))}}}

Divide both sides by n!

{{{3/(r!(n-r)!))}}}{{{""=""}}}{{{2/((r-1)!(n-r+1)!)}}}

{{{3((r-1)!(n-r+1)!^"")}}}{{{""=""}}}{{{2(r!(n-r)!^""))}}}

In the right side, write r! as r(r-1)!

{{{3((r-1)!(n-r+1)!^"")}}}{{{""=""}}}{{{2(r^""*(r-1)!)(n-r)!^""))}}}

Divide both sides by (r-1)!

{{{3(n-r+1)!}}}{{{""=""}}}{{{2r(n-r)!}}}

Write (n-r+1)! as (n-r+1)(n-r)!

{{{3(n-r+1)(n-r)!}}}{{{""=""}}}{{{2r(n-r)!}}}

Divide both sides by (n-r)!

{{{3(n-r+1)}}}{{{""=""}}}{{{2r}}}

{{{3n-3r+3}}}{{{""=""}}}{{{2r}}}

{{{3}}}{{{""=""}}}{{{5r-3n}}}

Looks better turned around:

{{{5r-3n}}}{{{""=""}}}{{{3}}}

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nC(r-2):nC(r-1) = 4:3 

Simplify the second equation:

{{{(nC(r-2))/(nC(r-1))}}}{{{""=""}}}{{{4/3}}}

{{{3(nC(r-2)^"")}}}{{{""=""}}}{{{4(nC(r-1)^"")}}}

{{{(3n!)/((r-2^"")!(n-(r-2)^"")!))}}}{{{""=""}}}{{{4n!/((r^""-1)!^""(n-(r-1)^"")!))}}}

{{{(3n!)/((r-2)!(n-r+2)!)}}}{{{""=""}}}{{{4n!/((r-1)!(n-r+1)!)}}}

Divide both sides by n!

{{{3/((r-2)!(n-r+2)!)}}}{{{""=""}}}{{{4/((r-1)!(n-r+1)!)}}}

{{{3((r-1)!(n-r+1)!^"")}}}{{{""=""}}}{{{4((r-2)!(n-r+2)!)}}}

In the left side, write (r-1)! as (r-1)(r-2)!

{{{3((r-1)(r-2)!(n-r+1)!^"")}}}{{{""=""}}}{{{4((r-2)!(n-r+2)!)}}}

Divide both sides by (r-2)!

{{{3(r-1)(n-r+1)!}}}{{{""=""}}}{{{4(n-r+2)!}}}

On the right side, write (n-r+2)! as (n-r+2)(n-r+1)!

{{{3(r-1)(n-r+1)!}}}{{{""=""}}}{{{4(n-r+2)(n-r+1)!}}}

Divide both sides by (n-r+1)!

{{{3(r-1)}}}{{{""=""}}}{{{4(n-r+2)}}}

{{{3r-3}}}{{{""=""}}}{{{4n-4r+8}}}

{{{7r-4n}}}{{{""=""}}}{{{11}}}


Now we solve the system of two equations for r and n

{{{system(5r-3n=3, 7r-4n=11)}}}

I'll let you solve that by elimination (addition). 

n = 34, r = 21

Checking in the first given proportion:

nCr:nC(r-1) = 2:3 , nCr-2:nCr-1 = 4:3

34C21:34C20 = 927983760:1391975640

Both those huge integers are divisible by the
huge integer 463991880

927983760/463991880 = 2 and 1391975640/463991880 = 3

So that checks.

Checking n = 34, r = 21 in the second given proportion:

nC(r-2):nC(r-1) = 4:3

34C19:34C20 = 1855967520:1391975640

Both those huge integers are divisible by the
huge integer 463991880

1855967520/463991880 = 4 and from above 1391975640/463991880 = 3

That also checks.

Edwin</pre></b>