answer:Let's break down the information given step by step: 1. Annie has 120 barrettes. 2. She has half as many scrunchies as barrettes. 3. She has 50 fewer bobby pins than three times the number of barrettes. 4. She has twice as many hairbands as the difference between the number of her bobby pins and scrunchies. First, we calculate the number of scrunchies: Since she has half as many scrunchies as barrettes, she has: 120 barrettes / 2 = 60 scrunchies Next, we calculate the number of bobby pins: She has 50 fewer bobby pins than three times the number of barrettes, so: (3 * 120 barrettes) - 50 = 360 - 50 = 310 bobby pins Now, we calculate the number of hairbands: The difference between the number of her bobby pins and scrunchies is: 310 bobby pins - 60 scrunchies = 250 She has twice as many hairbands as this difference, so: 2 * 250 = 500 hairbands To find the total number of hair decorations, we add up all the items: 120 barrettes + 60 scrunchies + 310 bobby pins + 500 hairbands = 990 total hair decorations To find the percentage of bobby pins, we divide the number of bobby pins by the total number of hair decorations and multiply by 100: (310 bobby pins / 990 total hair decorations) * 100 = 31.3131...% Rounded to the nearest percent, Annie has approximately boxed{31%} bobby pins in her total hair decorations collection.

answer:# Step-by-Step Solution Part 1: Finding the value of m Given the equation left(1+mxright)^{10}=a_{0}+a_{1}x+a_{2}x^{2}+ldots +a_{10}x^{10}, we know that the coefficient a_{5} corresponds to the term generated by selecting x^5 in the expansion, which can be represented using the binomial coefficient as {C}_{10}^{5}cdot m^{5}. Given a_{5}=-252, we can set up the equation: [ {C}_{10}^{5}cdot m^{5} = -252 ] The binomial coefficient {C}_{10}^{5} is calculated as: [ {C}_{10}^{5} = frac{10!}{5!5!} = 252 ] Substituting this into our equation gives: [ 252cdot m^{5} = -252 ] Solving for m gives: [ m^{5} = -1 implies m = -1 ] Thus, the value of m is boxed{m = -1}. Part 2: Calculating (a_{1}+a_{3}+a_{5}+a_{7}+a_{9})^{2}-(a_{0}+a_{2}+a_{4}+a_{6}+a_{8}+a_{10})^{2} Given the equation left(1-xright)^{10}=a_{0}+a_{1}x+a_{2}x^{2}+ldots +a_{10}x^{10}, we proceed as follows: 1. **Let x=1:** [ a_{0}+a_{1}+a_{2}+a_{3}+ldots +a_{9}+a_{10}=0 ] 2. **Let x=-1:** [ a_{0}-a_{1}+a_{2}-a_{3}+ldots -a_{9}+a_{10}=2^{10} ] Therefore, the sum of coefficients with alternating signs is 2^{10}. 3. **Calculating the desired expression:** [ (a_{1}+a_{3}+a_{5}+a_{7}+a_{9})^{2}-(a_{0}+a_{2}+a_{4}+a_{6}+a_{8}+a_{10})^{2} ] can be rewritten using the sums we found by setting x=1 and x=-1: [ = (a_{0}+a_{1}+a_{2}+a_{3}+ldots +a_{9}+a_{10})(-a_{0}+a_{1}-a_{2}+a_{3}+ldots +a_{9}+a_{10}) ] [ = 0 times (-2^{10}) = 0 ] Thus, the final answer is boxed{0}.

answer:**Analysis** This question tests the application of the formula for the distance between parallel lines and computational skills. By simplifying the equation of the lines, we can directly use the formula for the distance between parallel lines to solve the problem. **Solution** Given the line (l_{1}): (3x-3y+3=0), The distance between the two lines is: ( dfrac{left|3-1right|}{sqrt{3^{2}+3^{2}}} = dfrac{left|2right|}{3sqrt{2}} = dfrac{sqrt{2}}{3} ). Therefore, the answer is (boxed{dfrac{sqrt{2}}{3}}).

answer:From frac{z+1}{z-2}=1-3i, we get z+1=z-2-3iz+6i, which simplifies to 3iz=-3+6i. Therefore, z= frac{-3+6i}{3i}= frac{(-3+6i)(-i)}{-3i^{2}}= frac{6+3i}{3}=2+i. Thus, |z|= sqrt{2^{2}+1^{2}}= sqrt{5}. Hence, the correct choice is boxed{B}. This problem involves transforming the given equation, simplifying it using the algebraic operations of complex numbers, and then finding the magnitude of the complex number using its formula. It examines the operations of multiplication and division in the algebraic form of complex numbers and the method of finding the magnitude of a complex number, making it a basic question.