Question

There are n students in a school quiz club. The club need to select r students for a local competition, with r rounds (each student competes in one round each). Based on the number of students in the club, three of the students will not make the team and will have to be reserves. If the students are selected for the team and assigned a round of competition, there are 805 more ways to choose the team than if they only select the team members for now (ignoring the assignment of rounds). Use algebraic method to find the value of n.​

Answer 1

Given : There are n students in a school quiz club. The club need to select r students for a local competition, with r rounds (each student competes in one round each).

If the students are selected for the team and assigned a round of competition, there are 805 more ways to choose the team than if they only select the team members for now

To Find : the value of n

Solution:

Number of ways of selecting r students out of n  = $^nC_{r}$ = A

Number of ways of Assigning r students to r rounds = r!

A r! = A + 805

=> A( r!  - 1)  = 805

805 = 5 x 7 x 23

r!  - 1  can be  1  , 5  ,  7 ,  23 , 35  , 115 , 161 ,  805

only r!  - 1 = 5  or   r!  - 1 = 23 are possible

r = 3   or  r = 4

case 1 : r = 3

=> A = 161      ⁿC₃ = 161  not possible

case 2 : r = 4

=> A = 35      ⁿC₄ =  35  => n = 7

4 Students selected out of 7  in  35 ways

students are selected for the team and assigned a round of competition,  

= 35 x 4!  = 840

35 + 805 = 840

Hence n = 7   and r = 4

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