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question:w many distinct sequences of four letters can be made from the letters in PARADE if each sequence must begin with A, end with E, and no letter can appear in a sequence more than once?
answer:1. **Fix the first and last letters**: The sequence must start with A and end with E. Therefore, the sequence is of the form A _ _ E. 2. **Determine the choices for the remaining letters**: The remaining letters to choose from are P, A, R, D (noting that A has already been used once at the start and cannot be used again). 3. **Calculate the number of sequences**: - For the second position, we have three choices: P, R, D (since A is already used). - After choosing the second letter, we have two remaining choices for the third position (the two letters that were not chosen for the second position). - Therefore, the total number of distinct sequences is 3 cdot 2 = 6. Conclusion: The number of distinct sequences of four letters that can be formed under these conditions is boxed{6}.
question:What is the value of [2^{0+1+2} (2^0 + 2^1 + 2^2)?] textbf{(A) }1 qquad textbf{(B) }2 qquad textbf{(C) }3 qquad textbf{(D) }4 qquad textbf{(E) }5
answer:1. **Simplify the Exponent:** The expression given is 2^{0+1+2} - (2^0 + 2^1 + 2^2). First, simplify the exponent in the term 2^{0+1+2}: [ 0+1+2 = 3 ] Thus, the expression becomes: [ 2^3 - (2^0 + 2^1 + 2^2) ] 2. **Evaluate Powers of 2:** Calculate each power of 2: [ 2^3 = 8, quad 2^0 = 1, quad 2^1 = 2, quad 2^2 = 4 ] 3. **Substitute and Simplify:** Substitute these values into the expression: [ 8 - (1 + 2 + 4) ] Calculate the sum inside the parentheses: [ 1 + 2 + 4 = 7 ] Now subtract from 8: [ 8 - 7 = 1 ] 4. **Conclusion:** The value of the expression is 1, which is the result. Therefore, the corrected final answer is: [ 1 ] The final answer is boxed{textbf{(A) }1}
question:f ( f(x) = 5 2x ) and ( g(x) = x^2 + x + 1 ), find ( f(g(sqrt{3})) ).
answer:First, calculate ( g(sqrt{3}) ): g(sqrt{3}) = (sqrt{3})^2 + sqrt{3} + 1 = 3 + sqrt{3} + 1 = 4 + sqrt{3}. Next, find ( f(g(sqrt{3})) ): f(4 + sqrt{3}) = 5 - 2(4 + sqrt{3}) = 5 - (8 + 2sqrt{3}) = 5 - 8 - 2sqrt{3} = -3 - 2sqrt{3}. Thus, the solution is: boxed{-3 - 2sqrt{3}}.
question:A tangent line is drawn through the point (1, 0) to the parabola y=x^2+x+1. Among the options below, one of the tangent lines is: A: 2x+y+2=0 B: 3xy+3=0 C: x+y+1=0 D: xy+1=0
answer:First, we find the derivative of the parabola, which is y'=2x+1. Let's assume the coordinates of the tangent point are (x, y). Thus, the slope of the tangent line is 2x+1, and since y=x^2+x+1, the equation of the tangent line can be written as y-x^2-x-1=(2x+1)(x-x). Given that the point (-1, 0) lies on the tangent line, solving the equation yields x=0 or x=-2. When x=0, y=1; and when x=-2, y=3. This gives us two possible line equations, but after verification, option D is correct. Therefore, the correct choice is boxed{text{D}}.