1. **Problem Statement:** A computer lab has $12 + N$ identical computers to be distributed into 4 different rooms. Each room must receive at least 2 computers. Given $N=10$, find: a) The number of ways to distribute the identical computers. b) The number of ways if the computers are distinguishable. c) The number of ways if each room must receive exactly $3 + (N \bmod 3)$ computers. --- 2. **Formulas and Rules:** - For distributing identical items into distinct boxes with minimum constraints, use the stars and bars method. - For distinguishable items, use permutations and combinations. - The modulo operation $N \bmod 3$ gives the remainder when $N$ is divided by 3. --- 3. **Solution for (a):** - Total computers: $12 + N = 12 + 10 = 22$ - Each room gets at least 2 computers, so allocate 2 to each room first: $2 \times 4 = 8$ - Remaining computers to distribute: $22 - 8 = 14$ - Number of ways to distribute 14 identical computers into 4 rooms with no restriction: $$\binom{14 + 4 - 1}{4 - 1} = \binom{17}{3}$$ - Calculate $\binom{17}{3}$: $$\binom{17}{3} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = 680$$ --- 4. **Solution for (b):** - Computers are distinguishable, total $22$ computers. - Each room must have at least 2 computers. - Number of ways to distribute $22$ distinct computers into 4 distinct rooms with minimum 2 each: First, choose 2 computers for room 1: $\binom{22}{2}$ Then, choose 2 computers for room 2 from remaining 20: $\binom{20}{2}$ Then, choose 2 computers for room 3 from remaining 18: $\binom{18}{2}$ Room 4 gets the remaining $22 - 2 - 2 - 2 = 16$ computers. Number of ways to assign these 16 computers to room 4 is $\binom{16}{16} = 1$ (all remaining go there). Total ways: $$\binom{22}{2} \times \binom{20}{2} \times \binom{18}{2} = \frac{22 \times 21}{2} \times \frac{20 \times 19}{2} \times \frac{18 \times 17}{2}$$ Calculate stepwise: $$= 231 \times 190 \times 153 = 6719070$$ --- 5. **Solution for (c):** - $N \bmod 3 = 10 \bmod 3 = 1$ - Each room must receive exactly $3 + 1 = 4$ computers. - Total computers needed: $4 \times 4 = 16$ - But total computers available: $22$ - Since $16 \neq 22$, no ways to distribute exactly 4 computers per room. - Therefore, number of ways = 0. --- **Final answers:** a) $680$ b) $6719070$ c) $0$