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Home » Total lattice points: $2 + 2 = \boxed4$. - ToelettAPP

Total lattice points: $2 + 2 = \boxed4$. - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Thus, the smallest such integer is $\boxed{5}$.Question: An entomologist studies 5 species of bees, 3 species of wasps, and 2 species of beetles. If she observes one species per day for 10 days, how many distinct observation sequences are possible if species within each category are indistinguishable?

📌 Solution: The problem requires calculating the number of distinct permutations of 10 items where there are 5 identical bees, 3 identical wasps, and 2 identical beetles. The formula for multinomial coefficients applies:

📌 \frac{10!}{5! \cdot 3! \cdot 2!} = \frac{3628800}{120 \cdot 6 \cdot 2} = \frac{3628800}{1440} = 2520

📌 The number of distinct observation sequences is $\boxed{2520}$.

📌 Question: A science communicator creates a 10-part video series with 4 documentaries, 3 animations, and 3 interviews. If the order of video types matters but identical types are indistinct, how many unique sequences can be arranged?

📌 Solution: This is a multinomial permutation problem. The total sequences are given by:

📌 \frac{10!}{4! \cdot 3! \cdot 3!} = \frac{3628800}{24 \cdot 6 \cdot 6} = \frac{3628800}{864} = 4200

📌 The number of unique video sequences is $\boxed{4200}$.

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Also, $ f(ab) = 0 = 0 \cdot 0 = f(a)f(b) $, so the zero function works...
Now suppose $ f $ is not constant.
Let’s try additive functions satisfying both properties. From the addi...
f(a + b) = f(a) + f(b)
So $ f $ is additive. Also, from the multiplicative condition:
So $ f $ is both additive and multiplicative. These are very restricti...
Known result: The only functions $ f : \mathbb{R} \to \mathbb{R} $ tha...
$ f(x) = 0 $ for all $ x $
$ f(x) = x $ for all $ x $
Let’s verify both:

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