Respuesta :
Answer: 1344 committees can be formed of four people not from one class
Step-by-step explanation:
We are given:
Members in freshmen = 6
Members in sophomores = 6
Members in juniors = 6
To form a committee of four people not from one class, numerous combinations can occur. They are:
- When in a committee, at least one of them are from each class
When 2 from freshmen, 1 from sophomores, and 1 from junior. The possibility becomes [tex]^6C_2\times ^5C_1\times ^4C_1[/tex]
When 1 from freshmen, 2 from sophomores, and 1 from junior. The possibility becomes [tex]^6C_1\times ^5C_2\times ^4C_1[/tex]
When 1 from freshmen, 1 from sophomores, and 2 from junior. The possibility becomes [tex]^6C_1\times ^5C_1\times ^4C_2[/tex]
Total combinations: [tex](^6C_2\times ^5C_1\times ^4C_1)+(^6C_1\times ^5C_2\times ^4C_1)+(^6C_1\times ^5C_1\times ^4C_2)=(300+240+180)=720[/tex]
- If two members come from same class
When 2 from freshmen, 2 from sophomores, and 0 from junior. The possibility becomes [tex]^6C_2\times ^5C_2\times ^4C_0[/tex]
When 0 from freshmen, 2 from sophomores, and 2 from junior. The possibility becomes [tex]^6C_0\times ^5C_2\times ^4C_2[/tex]
When 2 from freshmen, 0 from sophomores, and 2 from junior. The possibility becomes [tex]^6C_2\times ^5C_0\times ^4C_2[/tex]
Total combinations: [tex](^6C_2\times ^5C_2\times ^4C_0)+(^6C_0\times ^5C_2\times ^4C_2)+(^6C_2\times ^5C_0\times ^4C_2)=(150+60+90)=300[/tex]
- If three members come from same class
When 3 from freshmen, 1 from sophomores, and 0 from junior. The possibility becomes [tex]^6C_3\times ^5C_1\times ^4C_0[/tex]
When 3 from freshmen, 0 from sophomores, and 1 from junior. The possibility becomes [tex]^6C_3\times ^5C_0\times ^4C_1[/tex]
When 0 from freshmen, 3 from sophomores, and 1 from junior. The possibility becomes [tex]^6C_0\times ^5C_3\times ^4C_1[/tex]
When 1 from freshmen, 3 from sophomores, and 0 from junior. The possibility becomes [tex]^6C_1\times ^5C_3\times ^4C_0[/tex]
When 0 from freshmen, 1 from sophomores, and 3 from junior. The possibility becomes [tex]^6C_0\times ^5C_1\times ^4C_3[/tex]
When 1 from freshmen, 0 from sophomores, and 3 from junior. The possibility becomes [tex]^6C_1\times ^5C_0\times ^4C_3[/tex]
Total combinations: [tex][(^6C_3\times ^5C_1\times ^4C_0)+(^6C_3\times ^5C_0\times ^4C_1)+(^6C_0\times ^5C_3\times ^4C_1)+(^6C_1\times ^5C_3\times ^4C_0)+(^6C_0\times ^5C_1\times ^4C_3)+(^6C_1\times ^5C_0\times ^4C_3)]=(100+80+40+60+20+24)=324[/tex]
Total number of committees that can be formed = [720 + 300 + 324] = 1344
Hence, 1344 committees can be formed of four people not from one class
