answer:To solve this problem, we need to consider the order in which the gloves can be worn. Since the two inner gloves are identical, they do not contribute to the sequence's diversity. The distinction comes from the outer gloves, which are specific to the left and right hands. First, consider wearing the inner gloves. Since they are identical, there is only 1 way to wear them, regardless of the order. Next, consider the outer gloves. There are 2 hands and 2 distinct gloves (left and right), so there are 2! = 2 times 1 = 2 ways to wear them. Therefore, the total number of different sequences for wearing the gloves is the product of the ways to wear the inner gloves and the ways to wear the outer gloves, which is 1 times 2 = 2. However, this calculation only considers one layer of gloves. Since the question implies wearing both inner and outer gloves, we need to reconsider the calculation. For each hand, you can either wear the inner glove first or the outer glove first. This gives us 2 choices per hand, and since there are 2 hands, the total number of sequences is 2^2 = 4. But, considering the requirement for the outer gloves to be worn correctly on the left and right hands, we realize the initial interpretation was incorrect. The correct interpretation should consider the combinations of wearing the inner gloves first or the outer gloves first, and then correctly wearing the outer gloves on the designated hands. Thus, for each hand, you have the choice of wearing the inner glove first or the outer glove first, leading to 2 choices per hand. Since there are 2 hands, this results in 2^2 = 4 ways to decide the order of inner and outer gloves. However, since the outer gloves are distinct and must be worn on the correct hand, this factor was already correctly accounted for in the initial calculation. Therefore, the correct answer, taking into account the distinct nature of the outer gloves and the identical nature of the inner gloves, along with the sequence of wearing them, is boxed{text{B: 6}}.

answer:To solve this problem, we start by recognizing that frac{1}{3^m} can be represented as frac{10^m}{3^m} = frac{10^m}{3^m}. We simplify this by noting that 10^m divided by 3^m simplifies to 3.overline{3} times 10^m = left(frac{10}{3}right)^m. We compute powers of frac{10}{3}, looking for an integer whose digits sum to more than 15. Here are the calculations: begin{align*} left(frac{10}{3}right)^1 &approx 3.333ldots left(frac{10}{3}right)^2 &approx 11.111ldots left(frac{10}{3}right)^3 &approx 37.037ldots left(frac{10}{3}right)^4 &approx 123.456ldots left(frac{10}{3}right)^5 &approx 411.522ldots end{align*} The sum of the digits of 411 is 4 + 1 + 1 = 6. We continue calculating: begin{align*} left(frac{10}{3}right)^6 &approx 1371.742ldots end{align*} The sum of the digits of 1371 is 1 + 3 + 7 + 1 = 12. We continue: begin{align*} left(frac{10}{3}right)^7 &approx 4572.473ldots end{align*} The sum of the digits of 4572 is 4 + 5 + 7 + 2 = 18. Thus, the smallest positive integer m such that the sum of the digits of frac{1}{3^m} that are to the right of the decimal point is greater than 15 is m = boxed{7}.

answer:# Step-by-Step Solution Part 1: Determining if 2x-3=0 is the k-shifted equation of 2x-1=0 1. Solve 2x-3=0: [ 2x = 3 implies x = frac{3}{2} ] 2. Solve 2x-1=0: [ 2x = 1 implies x = frac{1}{2} ] 3. Calculate the difference between the solutions: [ frac{3}{2} - frac{1}{2} = 1 ] Since the difference is 1, which is a positive integer, the equation 2x-3=0 is indeed the k-shifted equation of 2x-1=0. Therefore, the answer to part (1) is: boxed{text{yes}}. Part 2: Finding the value of n when 2x+m+n=0 is the "2-shifted equation" of 2x+m=0 1. Solve 2x+m+n=0: [ 2x = -m-n implies x = frac{-m-n}{2} ] 2. Solve 2x+m=0: [ 2x = -m implies x = frac{-m}{2} ] 3. Since it's a "2-shifted equation", we set up the equation: [ frac{-m-n}{2} - frac{-m}{2} = 2 ] 4. Solving for n: [ -n = 4 implies n = -4 ] Therefore, the value of n is: boxed{-4}. Part 3: Finding the value of 2b-2(c+3) when 5x+b=1 is the "3-shifted equation" of 5x+c=1 1. Solve 5x+b=1: [ 5x = 1-b implies x = frac{1-b}{5} ] 2. Solve 5x+c=1: [ 5x = 1-c implies x = frac{1-c}{5} ] 3. Since it's a "3-shifted equation", we set up the equation: [ frac{1-b}{5} - frac{1-c}{5} = 3 ] 4. Solving for b-c: [ b-c = -15 ] 5. Calculate 2b-2(c+3): [ 2b-2c-6 = 2(b-c)-6 = 2(-15)-6 = -30-6 = -36 ] Therefore, the value of 2b-2(c+3) is: boxed{-36}.

answer:Following the order of operations (PEMDAS/BODMAS): 1. Address any exponentiations first: [ 2^3 = 8 ] 2. Perform all divisions next: [ 12 div 3 = 4 ] 3. Complete the equation by performing the addition and then the subtraction: [ 5 + 4 - 8 ] First, add 5 and 4: [ 5 + 4 = 9 ] Finally, subtract 8 from 9: [ 9 - 8 = 1 ] The final answer to the problem is boxed{1}.