How many committees of 5 members each can be formed with 8 officials and 4 non - official members in the following cases:
each contains at least two non-official members

Answer

To solve the problem of forming committees of 5 members from 8 officials and 4 non-official members, ensuring that each committee contains at least 2 non-official members, we can follow these steps:

### Step 1: Determine Total Ways to Select 5 Members
We first calculate the total number of ways to select 5 members from the total of 12 members (8 officials + 4 non-officials) without any restrictions. This is given by the combination formula:

\[ \text{Total selections} = \binom{12}{5} \]

### Step 2: Calculate the Combinations
Using the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\):

\[ \binom{12}{5} = \frac{12!}{5!(12-5)!} = \frac{12!}{5! \cdot 7!} = 792 \]

### Step 3: Subtract Cases with Fewer than 2 Non-Official Members
Now we need to subtract the cases where there are fewer than 2 non-official members. This includes the cases where there are 0 non-official members and 1 non-official member.

#### Case 1: 0 Non-Official Members
If there are 0 non-official members, all 5 members must be officials. The number of ways to choose 5 officials from 8 is:

\[ \text{Selections with 0 non-officials} = \binom{8}{5} \cdot \binom{4}{0} \]

Calculating this:

\[ \binom{8}{5} = \frac{8!}{5! \cdot 3!} = 56 \]
\[ \binom{4}{0} = 1 \]

Thus, the total for this case is:

\[ 56 \cdot 1 = 56 \]

#### Case 2: 1 Non-Official Member
If there is 1 non-official member, then we need to choose 4 officials from the 8. The number of ways to do this is:

\[ \text{Selections with 1 non-official} = \binom{8}{4} \cdot \binom{4}{1} \]

Calculating this:

\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} = 70 \]
\[ \binom{4}{1} = 4 \]

Thus, the total for this case is:

\[ 70 \cdot 4 = 280 \]

### Step 4: Total Selections with Fewer than 2 Non-Official Members
Now we add the two cases together:

\[ \text{Total with fewer than 2 non-officials} = 56 + 280 = 336 \]

### Step 5: Calculate the Final Result
Finally, we subtract the total selections with fewer than 2 non-official members from the total selections:

\[ \text{Valid selections} = \text{Total selections} - \text{Total with fewer than 2 non-officials} \]
\[ \text{Valid selections} = 792 - 336 = 456 \]

### Conclusion
Thus, the total number of committees of 5 members each that can be formed with at least 2 non-official members is:

\[ \boxed{456} \]