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, two 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 the competition, there are 345 more ways to choose the team than if they only select the team members for now (ignoring the assignment of rounds).

Use an algebraic method to find the value of n.

Answer 1

Answer:

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Answer 2

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 345 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 + 345

=> A( r!  - 1)  = 345

345 = 3 x 5 x 23

r!  - 1  can be  1  , 3 , 5  ,  15 ,  23 , 69 ,  115 , 345

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

r = 2  , 3   or  4

case 1 : r = 2

=> A = 345      ⁿC₂ = 345  not possible

case 2 : r = 3

=> A = 69      ⁿC₃ = 69  not possible

case 3 : r = 4

=> A = 15      ⁿC₄ =  15  => n = 6

4 Students selected out of 6  in  15 ways

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

= 15 x 4!  = 360

15 + 345 = 360

Hence n = 6   and r = 4

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