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
Answers
Answer:
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}
n
C
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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