Question

find n and r if ncr-1 :ncr : ncr+1 = 14 : 8 : 3​

Answer 1

Step-by-step explanation:

find n and r if ncr-1 :ncr : ncr+1 = 14 : 8 : 3

Answer 2

The value of n is 10 and r is 7 in the given combination ratio expression.

Combination Calculation :

• Combinations are the count of the number of ways of selection of 'r' items from a given set of 'n' items ( each unique )
• It is denoted using subscript and superscript : $C^{n} _{r}$
• The formula for it is :

              $C_{r} ^{n} = (n!)/(n-r)!(r)!$

Using the formula in the expansion of the given ratio equation :

             $C_{r-1} ^{n} = (n!)/(n-r+1)!(r-1)!$

             $C_{r} ^{n} = (n!)/(n-r)!(r)!$

             $C_{r+1} ^{n} = (n!)/(n-r-1)!(r+1)!$

Equation 1: Comparing Ratios 1 and 2 :

Divide the quantities of count 1 and 2 to get the ratio of 14 : 8.

             $C_{r-1} ^{n} /C_{r} ^{n} = 14/8\\\\ ((n!)/(n-r+1)!(r-1)! )/( (n!)/(n-r)!(r)!)= 14/8\\\\ ((n-r)!(r)! )/( (n-r+1)!(r-1)!)= 14/8\\\\r/(n-r+1) = 14/8\\\\8r = 14n - 14r + 14\\\\22r - 14n = 14$

Equation 2: Comparing Ratios 2 and 3 :

Divide the quantities of count 2 and 3 to get the ratio of 8 : 3.

             $C_{r} ^{n} /C_{r+1} ^{n} = 8/3\\\\ ((n!)/(n-r)!(r)! )/( (n!)/(n-r-1)!(r+1)!)= 8/3\\\\ ((n-r-1)!(r+1)! )/( (n-r)!(r)!)= 8/3\\\\(r+1)/(n-r) = 8/3\\\\3r+3 = 8n - 8r \\\\8n- 11r = 3$

Solving the two equations and 2 variables system using the elimination method :

We do ( Equation1 + 2*Equation2 ) to eliminate the variable r.

          $22r - 14n + 2(8n-11r) = 14 +(3*2)\\22r - 14n + 16n-22r = 14 + 6\\2n = 20\\n=10$

Using the value of 'n' to find the value of 'r' :

         $22r - 14(10) = 14\\22r = 14 + 140\\22r = 154\\r = 7$

Hence, the value of n and r is 10 and 7 respectively.

           

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