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question:You have 7 red shirts, 6 green shirts, 5 blue shirts, 6 pairs of pants, 6 green hats, 7 red hats, and 5 blue hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?
answer:Consider three cases: bullet Case 1: the person wears a red shirt and either a green or blue hat. There are 7 options for red shirts, 6 options for pants, 6 options for green hats, and 5 options for blue hats, making a total of 7 times 6 times (6+5) = 7 times 6 times 11 = 462 outfits. bullet Case 2: the person wears a green shirt and either a red or blue hat. There are 6 green shirts, 6 options for pants, 7 options for red hats, and 5 options for blue hats, making a total of 6 times 6 times (7+5) = 6 times 6 times 12 = 432 outfits. bullet Case 3: the person wears a blue shirt and either a green or red hat. There are 5 blue shirts, 6 options for pants, 6 options for green hats, and 7 options for red hats, making a total of 5 times 6 times (6+7) = 5 times 6 times 13 = 390 outfits. Summing up the three cases, we have a total possibility of 462 + 432 + 390 = boxed{1284} outfits.
question:There are ( x ) terms in the algebraic expression ( x^{6} + x^{6} + x^{6} + ldots + x^{6} ) and its sum is ( x^{b} ). Find ( b ).
answer:1. The given expression is x^6 + x^6 + x^6 + ldots + x^6 where there are x terms of x^6. 2. We can represent the sum of this expression as: [ S = x cdot x^6 ] since there are x terms each contributing a x^6 to the sum. 3. Using the properties of exponents, we know that: [ x cdot x^6 = x^1 cdot x^6 = x^{1+6} = x^7 ] 4. We are given that this sum is equivalent to x^b. Therefore, we can write: [ x^7 = x^b ] 5. Because the bases are the same, we can equate the exponents: [ 7 = b ] 6. Thus, the value of b is: [ boxed{7} ]
question:The negation of the proposition For all x in mathbb{R}, x^2 2x + 4 leq 0 is ( ) A: For all x in mathbb{R}, x^2 2x + 4 geq 0 B: There exists an x in mathbb{R}, x^2 2x + 4 > 0 C: For all x notin mathbb{R}, x^2 2x + 4 leq 0 D: There exists an x notin mathbb{R}, x^2 2x + 4 > 0
answer:Given the proposition "For all x in mathbb{R}, x^2 - 2x + 4 leq 0", the negation of this proposition is "There exists an x in mathbb{R}, x^2 - 2x + 4 > 0" Therefore, the correct option is B. **Analysis:** This problem involves a universal proposition, whose negation is an existential proposition. The negation of the universal proposition can be written according to the form of negating a universal proposition. boxed{text{B}}
question:Find 97^{1} pmod{98}, as a residue modulo 98. (Give an answer between 0 and 97, inclusive.)
answer:To find 97^{-1} pmod{98}, we observe: 1. Calculate 97^2 pmod{98}: [ 97^2 = 9409 ] [ 9409 mod 98 = (98 times 96 + 1) mod 98 = 1 ] So, 97^2 equiv 1 pmod{98}. 2. Since 97^2 equiv 1 pmod{98}, it implies that 97 cdot 97 equiv 1 pmod{98}. Thus, 97 is its own inverse modulo 98. Therefore, 97^{-1} equiv boxed{97} pmod{98}.