<u>Answer:</u> 1344 committees can be formed of four people not from one class
<u>Step-by-step explanation:</u>
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 ![^6C_2\times ^5C_1\times ^4C_1](https://tex.z-dn.net/?f=%5E6C_2%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_1)
When 1 from freshmen, 2 from sophomores, and 1 from junior. The possibility becomes ![^6C_1\times ^5C_2\times ^4C_1](https://tex.z-dn.net/?f=%5E6C_1%5Ctimes%20%5E5C_2%5Ctimes%20%5E4C_1)
When 1 from freshmen, 1 from sophomores, and 2 from junior. The possibility becomes ![^6C_1\times ^5C_1\times ^4C_2](https://tex.z-dn.net/?f=%5E6C_1%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_2)
Total combinations: ![(^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](https://tex.z-dn.net/?f=%28%5E6C_2%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_1%29%2B%28%5E6C_1%5Ctimes%20%5E5C_2%5Ctimes%20%5E4C_1%29%2B%28%5E6C_1%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_2%29%3D%28300%2B240%2B180%29%3D720)
- If two members come from same class
When 2 from freshmen, 2 from sophomores, and 0 from junior. The possibility becomes ![^6C_2\times ^5C_2\times ^4C_0](https://tex.z-dn.net/?f=%5E6C_2%5Ctimes%20%5E5C_2%5Ctimes%20%5E4C_0)
When 0 from freshmen, 2 from sophomores, and 2 from junior. The possibility becomes ![^6C_0\times ^5C_2\times ^4C_2](https://tex.z-dn.net/?f=%5E6C_0%5Ctimes%20%5E5C_2%5Ctimes%20%5E4C_2)
When 2 from freshmen, 0 from sophomores, and 2 from junior. The possibility becomes ![^6C_2\times ^5C_0\times ^4C_2](https://tex.z-dn.net/?f=%5E6C_2%5Ctimes%20%5E5C_0%5Ctimes%20%5E4C_2)
Total combinations: ![(^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](https://tex.z-dn.net/?f=%28%5E6C_2%5Ctimes%20%5E5C_2%5Ctimes%20%5E4C_0%29%2B%28%5E6C_0%5Ctimes%20%5E5C_2%5Ctimes%20%5E4C_2%29%2B%28%5E6C_2%5Ctimes%20%5E5C_0%5Ctimes%20%5E4C_2%29%3D%28150%2B60%2B90%29%3D300)
- If three members come from same class
When 3 from freshmen, 1 from sophomores, and 0 from junior. The possibility becomes ![^6C_3\times ^5C_1\times ^4C_0](https://tex.z-dn.net/?f=%5E6C_3%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_0)
When 3 from freshmen, 0 from sophomores, and 1 from junior. The possibility becomes ![^6C_3\times ^5C_0\times ^4C_1](https://tex.z-dn.net/?f=%5E6C_3%5Ctimes%20%5E5C_0%5Ctimes%20%5E4C_1)
When 0 from freshmen, 3 from sophomores, and 1 from junior. The possibility becomes ![^6C_0\times ^5C_3\times ^4C_1](https://tex.z-dn.net/?f=%5E6C_0%5Ctimes%20%5E5C_3%5Ctimes%20%5E4C_1)
When 1 from freshmen, 3 from sophomores, and 0 from junior. The possibility becomes ![^6C_1\times ^5C_3\times ^4C_0](https://tex.z-dn.net/?f=%5E6C_1%5Ctimes%20%5E5C_3%5Ctimes%20%5E4C_0)
When 0 from freshmen, 1 from sophomores, and 3 from junior. The possibility becomes ![^6C_0\times ^5C_1\times ^4C_3](https://tex.z-dn.net/?f=%5E6C_0%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_3)
When 1 from freshmen, 0 from sophomores, and 3 from junior. The possibility becomes ![^6C_1\times ^5C_0\times ^4C_3](https://tex.z-dn.net/?f=%5E6C_1%5Ctimes%20%5E5C_0%5Ctimes%20%5E4C_3)
Total combinations: ![[(^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](https://tex.z-dn.net/?f=%5B%28%5E6C_3%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_0%29%2B%28%5E6C_3%5Ctimes%20%5E5C_0%5Ctimes%20%5E4C_1%29%2B%28%5E6C_0%5Ctimes%20%5E5C_3%5Ctimes%20%5E4C_1%29%2B%28%5E6C_1%5Ctimes%20%5E5C_3%5Ctimes%20%5E4C_0%29%2B%28%5E6C_0%5Ctimes%20%5E5C_1%5Ctimes%20%5E4C_3%29%2B%28%5E6C_1%5Ctimes%20%5E5C_0%5Ctimes%20%5E4C_3%29%5D%3D%28100%2B80%2B40%2B60%2B20%2B24%29%3D324)
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