Q. Suppose that M is the product of k distinct primes. Find the number of ways to write N as the product of positive integers(>1), where the order of terms does not matter.

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Q. How many ways are there to place 7 differently colored toys into 5 identical urns if the urns can be empty? Note that all balls have to be used.

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Q. Suppose, there are 7 of your friends who want to eat pizza (8 distinct people in total). You order a 16-cut pizza (16 identical slices). How many distributions of pizza slices are there if each person gets at least one slice of pizza?

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Q. The linear system Cx = d is known as _________ if d! = 0.

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Q. Every linear equation determines a _______ in n-dimensional space for n variables.

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Q. Determine all possibilities for the number of solutions of the system of 7 equations in 5 unknowns and it has x₁ = 0, x₂ = −6, and x₃ = 4 as a solution.

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Q. Determine all possibilities for the solution set of the homogeneous system that has y₁ = 6, y₂ = −4, y₃ = 0 as a solution.

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Q. Determine all possibilities for the solution set of the homogeneous system of 5 equations in 3 unknowns and the rank of the system is 3.

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Q. Determine all possibilities for the solution set of a homogeneous system that has y₁ = 5, y₂ = −3, y₃ = 2 as a solution.

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Q. Determine all possibilities for the solution set of the system of 2 equations in 3 unknowns that has x₁ = 4, x₂ = −7, x₃ = 0 as a solution.

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Q. Determine all possibilities for the solution set of a homogeneous system of 4 equations in 4 unknowns.

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Q. Determine all possibilities for the solution set of a homogeneous system of 6 equations in 5 unknowns.

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Q. Determine all possibilities for the solution set of a homogeneous system of 5 equations in 4 unknowns and the rank of the system is 3.

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Q. Determine the number of derangements of (2, 4, 6, 1, 3, 5) that end with integer 2, 4 and 6 in some order?

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Q. A nursery teacher has 5 pencil boxes to give out to her five students. Determine the probability that at least one student gets their name tag?

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Q. Farhan has received 9 gifts from 9 different people. In how many ways can Farhan receives the gifts such that no one gives him real gifts?

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Q. There are 7 groups in a picnic who has brought their own lunch box, and then the 7 lunch box are exchanged within those groups. Determine the number of ways that they can exchange the lunch box such that none of them can get their own.

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Q. Computational complexity of derangements is of __________

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Q. There are 5 different-colored boxes in a room each with a distinct cover. Find out the number of ways so that these covers can be put on the boxes such that none of the boxes can have right covers on it? (Assume that all the covers must be on the boxes).

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Q. A postman can put 12 letters into their respective envelopes such that exactly 5 will go into the right envelope. Find the number of ways of doing this work.

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