1. **Problem:** You want to arrange 12 canned goods in a row: 3 identical meat loaf cans, 4 identical tomato sauce cans, 2 identical sardine cans, and 3 identical corned beef cans. How many different ways can you display these goods? 2. **Formula:** When arranging $n$ items where there are groups of identical items, the number of distinct arrangements is given by the multinomial formula: $$\frac{n!}{n_1! \times n_2! \times \cdots \times n_k!}$$ where $n$ is the total number of items, and $n_1, n_2, \ldots, n_k$ are the counts of identical items in each group. 3. **Apply the formula:** Here, $n = 12$ (total cans), with groups: - Meat loaf: $3$ identical cans - Tomato sauce: $4$ identical cans - Sardines: $2$ identical cans - Corned beef: $3$ identical cans So the number of ways is: $$\frac{12!}{3! \times 4! \times 2! \times 3!}$$ 4. **Calculate factorials:** - $12! = 479001600$ - $3! = 6$ - $4! = 24$ - $2! = 2$ - $3! = 6$ 5. **Compute denominator:** $$3! \times 4! \times 2! \times 3! = 6 \times 24 \times 2 \times 6 = 1728$$ 6. **Final calculation:** $$\frac{479001600}{1728} = 277200$$ **Answer:** There are **277200** different ways to arrange the canned goods on the shelf.