All Questions ⁺⁰237643 Questions

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2. If |𝜓⟩ = |𝜙1⟩ + |𝜙2⟩ and |𝜒⟩ = |𝜙1⟩ − |𝜙2⟩ , Show that ⟨𝜓|𝜓⟩ + ⟨𝜒|𝜒⟩ = 2⟨𝜙1|𝜙1⟩ + 2⟨𝜙2|𝜙2⟩

2. If |𝜓⟩ = |𝜙1⟩ + |𝜙2⟩ and |𝜒⟩ = |𝜙1⟩ − |𝜙2⟩ , Show that ⟨𝜓|𝜓⟩ + ⟨𝜒|𝜒⟩ = 2⟨𝜙1|𝜙1⟩ + 2⟨𝜙2|𝜙2⟩

Guest Jan 12, 2023

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Distributions

(a) In how many ways can you distribute 4 distinguishable balls among 3 indistinguishable boxes?

(b) In how many ways can you distribute 4 indistinguishable balls among 5 distinguishable boxes?

Guest Jan 12, 2023

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195

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Binomial Theorem

(a) Simplify (n/k)/(n/(k - 1))

(b) For some positive integer n, the expansion of (1 + x)^n has three consecutive coefficients a, b, c that satisfy 1:8:28. What must n be?

Guest Jan 12, 2023

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204

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Counting

Find the number of positive integers that satisfy the following conditions:
* Each digit is a 1 or a 2 or a 3
* The sum of the digits is 10

Guest Jan 12, 2023

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169

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Probability

A bag contains five white balls and three black balls. Your goal is to draw two white balls.

(a) You simultaneously draw two balls at random. What is the probability that they are both white?
(b) You simultaneously draw two balls read more ..

Guest Jan 12, 2023

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199

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help math

Let n be a positive integer. Let k be a positive integer.

(a) Using a counting argument, show that C(n - 1, k - 1) + C(n - 1, k) = C(n,k).
read more ..

Guest Jan 12, 2023

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158

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Expected value

The grid below is made up of line segments, like the line segment in red.

There are a number of paths that go from A to B in the grid, where every step goes to the right or up. If we choose a line segment at random, then read more ..

Guest Jan 12, 2023

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194

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Function

The function f(n) satisfies
binom(2n)n - binom(2n + 1)(2n) = f(n) *binom(2n + 1)(n + 1)
for all positive integers n. Find f(n).

Guest Jan 12, 2023

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176

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Counting arrangements

Find the number of ways of arranging the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 in a row so that the product of any two adjacent numbers is even or a mutliple of 3

Guest Jan 12, 2023

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181

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Absolute value

Find all solutions of the equation
|x^2 - 30x - 1| + |x^2 - 30x + 29| = 30.

Guest Jan 12, 2023

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167

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Angle bisectors

Let ABC be a triangle, and let its angle bisectors be AD, BE, and CF, which intersect at I. If DI = 3, BD = 4, and BI = 8, then find the area of triangle ABC.

Guest Jan 12, 2023

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168

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help counting

In how many ways can the numbers 1 through 5 be entered once each into the five boxes below so that all the given inequalities are true?

Guest Jan 12, 2023

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155

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plz help

A stick is broken at two points, at random. If the length of the stick is 6, then what is the probability that all three resulting pieces are shorter than 3 units?

Guest Jan 12, 2023

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164

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was ist rad

Guest Jan 12, 2023

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Geometry

In triangle PQR, M is the midpoint of PQ. Let X be the point on QR such that PX bisects angle QPR, and let the perpendicular bisector of PQ intersect AX at Y. If PQ = 36, PR = 22, QR = 26, and MY = 8, then find the area read more ..

Guest Jan 12, 2023

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