💡Okay, so I need to figure out how many distinct arrangements there are for the letters in the word "balloon." Hmm, let me think about this step by step.First, I know that when dealing with arrangements of letters, especially when there are repeating letters, I need to use permutations. But since some letters repeat, I can't just use the simple factorial of the number of letters. I remember there's a formula for permutations of a multiset, which accounts for repeated elements.Let me recall the formula: the number of distinct arrangements is equal to the factorial of the total number of items divided by the product of the factorials of the counts of each repeated item. So, in mathematical terms, it's n! divided by (n1! * n2! * ... * nk!), where n is the total number of items, and n1, n2, ..., nk are the counts of each repeated item.Alright, so applying this to "balloon." First, I need to know how many letters there are in total. Let's count them: B, A, L, L, O, O, N. That's 7 letters in total.Now, I need to identify which letters are repeating and how many times they repeat. Looking at "balloon," I see that the letter "L" appears twice and the letter "O" also appears twice. The other letters—B, A, and N—appear only once each.So, according to the formula, the number of distinct arrangements should be 7! divided by (2! for the two Ls and 2! for the two Os). Let me write that out:Number of arrangements = 7! / (2! * 2!)Now, calculating the factorials:7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 50402! = 2 × 1 = 2So, plugging those numbers back into the equation:Number of arrangements = 5040 / (2 * 2) = 5040 / 4 = 1260Wait, that seems straightforward. But let me double-check to make sure I didn't miss anything. Did I account for all the repeated letters? Yes, "L" and "O" each repeat twice, and I divided by 2! for each. The other letters only appear once, so they don't affect the denominator.Is there any other way this could be approached? Maybe by considering the positions one by one, but that seems more complicated. The formula I used should suffice.Just to be thorough, let me think about what this number represents. It's the total number of unique ways to arrange the letters in "balloon," considering that swapping the two Ls or the two Os doesn't create a new arrangement. So, by dividing by 2! for each set of duplicates, I'm effectively removing those redundant arrangements.Yeah, I think that makes sense. So, my final answer should be 1260 distinct arrangements.

💡First, identify the quantities of each item Rebecca has:- Eggs: 16- Marbles: 3- Rubber Bands: 5To create groups with an equal number of eggs, marbles, and rubber bands, we need to find a common factor that divides all three quantities evenly.1. **Determine Possible Number of Groups:** - The number of groups must be a factor of the smallest quantity, which is the marbles (3). - Possible number of groups: 1 or 3.2. **Check for 3 Groups:** - **Eggs:** 16 ÷ 3 = 5 with a remainder of 1. Eggs do not divide evenly. - **Marbles:** 3 ÷ 3 = 1. Marbles divide evenly. - **Rubber Bands:** 5 ÷ 3 = 1 with a remainder of 2. Rubber bands do not divide evenly. Since eggs and rubber bands do not divide evenly into 3 groups, 3 groups are not possible.3. **Check for 1 Group:** - **Eggs:** 16 ÷ 1 = 16. Eggs divide evenly. - **Marbles:** 3 ÷ 1 = 3. Marbles divide evenly. - **Rubber Bands:** 5 ÷ 1 = 5. Rubber bands divide evenly. All items divide evenly into 1 group.**Conclusion:** Rebecca can create **1 group** with all the eggs, marbles, and rubber bands.

💡First, calculate the total number of pizza slices by adding the slices from both pizzas.Next, determine the total number of slices consumed by both Phil and Andre.Subtract the total consumed slices from the total available slices to find out how many slices are left.Finally, divide the remaining slices equally between Phil and Andre to find out how many slices each person gets.

💡First, I need to determine the constant of proportionality ( k ) using the given point ( (2, 3) ) on the graph of the inverse proportion function ( y = frac{k}{x} ).Substituting ( x = 2 ) and ( y = 3 ) into the equation:[3 = frac{k}{2}]Solving for ( k ):[k = 2 times 3 = 6]Now, I need to check which of the given points satisfies the condition ( xy = k ) with ( k = 6 ).- **Option A: ( (-2, 3) )** [ xy = -2 times 3 = -6 quad (text{Not equal to } 6) ]- **Option B: ( (1, 5) )** [ xy = 1 times 5 = 5 quad (text{Not equal to } 6) ]- **Option C: ( (1, 6) )** [ xy = 1 times 6 = 6 quad (text{Equal to } 6) ]- **Option D: ( (1, -6) )** [ xy = 1 times -6 = -6 quad (text{Not equal to } 6) ]Only option C satisfies the condition ( xy = 6 ).

💡First, I need to subtract the second polynomial from the first one. I'll start by aligning the like terms to ensure each corresponding term is properly subtracted.Looking at the (x^6) terms, I have (2x^6) in the first polynomial and (x^6) in the second. Subtracting these gives (2x^6 - x^6 = x^6).Next, for the (x^5) terms, there's (3x^5) in the first polynomial and (4x^5) in the second. Subtracting these results in (3x^5 - 4x^5 = -x^5).Moving on to the (x^4) terms, I have (2x^4) in the first polynomial and (x^4) in the second. Subtracting these gives (2x^4 - x^4 = x^4).For the (x^3) term, the first polynomial doesn't have an (x^3) term, so it's effectively (0x^3). The second polynomial has (-x^3). Subtracting these gives (0x^3 - (-x^3) = x^3).For the (x) term, the first polynomial has (x) and the second doesn't have an (x) term, so it's effectively (0x). Subtracting these gives (x - 0x = x).Finally, for the constant terms, I have (15) in the first polynomial and (20) in the second. Subtracting these results in (15 - 20 = -5).Putting all these simplified terms together, the final simplified expression is (x^6 - x^5 + x^4 + x^3 + x - 5).

💡First, I recognize that the pass rate of the product is 95%. This means that out of all the products produced, 95% meet the basic quality standards.Next, I understand that among these qualified products, 20% are classified as first-class. This indicates that only a portion of the already qualified products achieve the highest quality level.To find the overall rate of first-class products, I need to calculate the percentage of first-class products relative to the total number of products produced. This involves multiplying the pass rate by the rate of first-class products among the qualified ones.So, the calculation would be:95% (pass rate) multiplied by 20% (first-class rate among qualified products) equals 19%.Therefore, the rate of first-class products for this product is 19%.