There are 14 juniors and 16 seniors in a chess club. a) From the 30 members, how many ways are there to arrange 5 members of the club in a line? b) How many ways are there to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line? 0 c) If the club sends 2 juniors and 2 seniors to the tournament, how many possible groupings are there? d) If the club sends either 4 juniors or 4 seniors, how many possible groupings are there?
Accepted Solution
A:
Answer:No. of juniors = 14No. of seniors = 16Total students = 30 A) From the 30 members, how many ways are there to arrange 5 members of the club in a line?Since we are asked about arrangement so we will use permutation Formula : [tex]^nP_r=\frac{n!}{(n-r)!}[/tex]n = 30 r = 5[tex]^{30}P_5=\frac{30!}{(30-5)!}[/tex][tex]^{30}P_5=17100720[/tex]So, From the 30 members, there are 17100720 ways to arrange 5 members of the club in a line?B) How many ways are there to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line?Out of 16 seniors 2 will be selectedSo, 3 places are vacantRemaining students = 30-2 = 28So, out of 28 students 3 students will be selectedNo. of ways = [tex]^{16}P_2 \times ^{28}P_3[/tex]No. of ways = [tex]\frac{16!}{(16-2)!}\times\frac{28!}{(28-3)!}[/tex] = [tex]4717440[/tex] There are 4717440 ways to arrange 5 members of the club in a line if there must be a senior at the beginning of the line and at the end of the line.C)If the club sends 2 juniors and 2 seniors to the tournament, how many possible groupings are there?Since we are not asked about arrangement so we will use combinationOut of 16 seniors 2 will be selectedOut of 14 juniors 2 will be selectedFormula : [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]So, No. of possible groupings = [tex]^{16}C_2 \times ^{14}C_2[/tex] = [tex]\frac{16!}{2!(16-2)!} \times \frac{14!}{2!(14-2)!}[/tex] = [tex]10920[/tex]If the club sends 2 juniors and 2 seniors to the tournament, there are 10920 possible groupings D) If the club sends either 4 juniors or 4 seniors, how many possible groupings are there?Out of 16 seniors 4 will be selectedorOut of 14 juniors 4 will be selectedSo, No. of possible groupings = [tex]^{16}C_4 + ^{14}C_4[/tex] = [tex]\frac{16!}{4!(16-4)!} + \frac{14!}{4!(14-4)!}[/tex] = [tex]2821[/tex]So,If the club sends either 4 juniors or 4 seniors, there are 2821 possible groupings .