Question

If ncr:ncr+1:ncr+2=1:3:5 then find the veule of n and

Answer 1

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♦♦ Combination Functions ♦♦

$^nC_r : \ ^nC_{r+1} : \ ^nC_{r+2} = 1 : 3 : 5$

→ C ( n , r ) : C( n , r + 1 ) = 1 : 3

[tex]=\ \textgreater \ \frac{n!}{r!(n-r)!} : \frac{n!}{(r+1)r!(n-r-1)!} = 1 : 3 \\ \\ =\ \textgreater \ 3( r + 1 ) = ( n - r ) \\ =\ \textgreater \ 4r = n - 3 [/tex]

→ C( n , r + 1 ) : C( n , r + 2 ) = 3 : 5

[tex] \frac{n!}{(r+1)!(n-r-1)!} : \frac{n!}{(r+2)!(n-r-2)!} = 3 : 5 \\ \\ =\ \textgreater \ 5( r + 2 ) = 3( n - r - 1 ) \\ =\ \textgreater \ 8r = 3n - 13 \\ =\ \textgreater \ 2n - 6 = 3n - 13 \\ =\ \textgreater \ n = 7[/tex]

Note : We use [ 4r = n - 3 ] and substitute it into Second Simplification to get :  n = 7 as the result