Answer: 0.04848
Step-by-step explanation:
Given : There are 4 sophomores, 4 juniors and 3 seniors.
Total choices = 4+4+3 = 11
Number of people needed to be chosen for committee = 4
Number of ways to select 3 people out of 11 (without replacement)= [tex]^{11}P_4[/tex]
[tex]=\dfrac{11!}{(11-4)!}\ \ \ \ [^nP_r=\dfrac{n!}{(n-r)!}][/tex]
[tex]=\dfrac{11\times10\times9\times8\times7!}{7!}\\\\= 7920[/tex]
Number of ways to select at least one representative from all three groups of students = [tex]^4P_2\times ^4P_1\times ^3P_1+^4P_1\times ^4P_2\times ^3P_1+^4P_1\times ^4P_1\times ^3P_2[/tex]
[tex]=\dfrac{4!}{2!}\times4\times3+4\dfrac{4!}{2!}\times3+4\times4\times\dfrac{3!}{2!}\\\\= 144+144+96=384[/tex]
Required probability = [tex]\dfrac{384}{7920}=0.04848[/tex]
Hence, the probability the committee thus formed will have at least one representative from all three groups of students = 0.04848
