1. **Problem statement:** There are 4 different Mathematics books and 5 different Filipino books. We want to find the number of ways to arrange these books on a shelf such that books of the same subject are placed together. 2. **Formula and explanation:** When items are grouped and each group must stay together, treat each group as a single item first, then multiply by the permutations within each group. 3. **Step-by-step solution:** - Treat the Mathematics books as one group and the Filipino books as another group. So, we have 2 groups to arrange. - Number of ways to arrange the 2 groups: $2! = 2$ - Number of ways to arrange the 4 Mathematics books within their group: $4! = 24$ - Number of ways to arrange the 5 Filipino books within their group: $5! = 120$ 4. **Calculate total arrangements:** $$\text{Total ways} = 2! \times 4! \times 5! = 2 \times 24 \times 120 = 5760$$ 5. **Final answer:** There are $5760$ ways to arrange the books with the given condition. --- 1. **Problem statement:** In a gathering with 20 guests, each guest shakes hands with every other guest. Find the total number of handshakes. 2. **Formula and explanation:** Each handshake involves 2 guests. The total number of handshakes is the number of unique pairs of guests. 3. **Step-by-step solution:** - Number of ways to choose 2 guests out of 20: $\binom{20}{2} = \frac{20 \times 19}{2} = 190$ 4. **Final answer:** There will be $190$ handshakes in total.