Question

Notify Me Bookmark

You toss two six-sided dice. What is the total number of ways in which you can obtain the following? (a) a 7 (b) a 9 What If? If you now rolled three six-sided dice, what would be the number of ways in which you can obtain the following? (c) a 12 (d) a 9

You toss two six-sided dice. What is the total number of ways in which you can obtain the following? (a) a 7 (b) a 9 What If? If you now rolled three six-sided dice, what would be the number of ways in which you can obtain the following? (c) a 12 (d) a 9

Image text
You toss two six-sided dice. What is the total number of ways in which you can obtain the following? (a) a 7 ◻ (b) a 9 ◻ What If? If you now rolled three six-sided dice, what would be the number of ways in which you can obtain the following? (c) a 12 ◻ (d) a 9 ◻

Detailed Answer

0 0

Please enter your email here. You will get email when this question is answered by our tutor.

Doubtrix Logo

Problem is not yet solved!

Click on Notify me to get email when question is answered.

Join Doubtrix with 7-days free trial

  • Icon Correct & Verified Answers
  • Icon No AI-Generated Answers
  • Icon Video Solution for Difficult Questions
  • Icon Work With Experts to Reach at Correct Answers
Notify me Login Sign Up

Similar/Related Questions

For phonon gas, we have introduced two models to describe its statistical properties. Namely they are the Einstein model and the Debye model. We have learnt that for non-interacting systems, the statistical properties (i.e. the partition function) are mainly controlled by the dispersion relations and the density of states. The latter also strongly depends on the dimension of the system. (a) Discuss about the dispersion relations and the density of states of the two phonon models. Make plots to show their differences. (b) For phonons, the total number of modes are fixed. Note this does not imply the total number of phonons is fixed. Each mode is treated as a quantum harmonic oscillator, so it is a state that can be occupied by any integer number of phonons (since phonons are bosons). We are thus looking at partition functions of the "grand canonical ensemble" where number of phonons are not fixed. Write down the partition function of the Einstein model and the Debye model in three dimensions, and explain why we do not have a chemical potential in the partition function. (c) Derive the specific heat capacity for the Debye model from the partition function above. You will encounter an integration that is difficult to do, and you can leave it so. Try to do a change of variable for the integration, so the integrand is dimensionless. Explain why the integration is still dependent on temperature. (a mistake in the lecture notes was corrected). (d) Can you also derive the specific heat capacity in two-dimension using the Debye model? Note the total number of modes also depend on the spatial dimension, which will affect the Debye frequency.

For phonon gas, we have introduced two models to describe its statistical properties. Namely they are the Einstein model and the Debye model. We have learnt that for non-interacting systems, the statistical properties (i.e. the partition function) are mainly controlled by the dispersion relations and the density of states. The latter also strongly depends on the dimension of the system. (a) Discuss about the dispersion relations and the density of states of the two phonon models. Make plots to show their differences. (b) For phonons, the total number of modes are fixed. Note this does not imply the total number of phonons is fixed. Each mode is treated as a quantum harmonic oscillator, so it is a state that can be occupied by any integer number of phonons (since phonons are bosons). We are thus looking at partition functions of the "grand canonical ensemble" where number of phonons are not fixed. Write down the partition function of the Einstein model and the Debye model in three dimensions, and explain why we do not have a chemical potential in the partition function. (c) Derive the specific heat capacity for the Debye model from the partition function above. You will encounter an integration that is difficult to do, and you can leave it so. Try to do a change of variable for the integration, so the integrand is dimensionless. Explain why the integration is still dependent on temperature. (a mistake in the lecture notes was corrected). (d) Can you also derive the specific heat capacity in two-dimension using the Debye model? Note the total number of modes also depend on the spatial dimension, which will affect the Debye frequency.

Solution
0 0

Children playing in a playground on the flat roof of a city school lose their ball to the parking lot below. One of the teachers kicks the ball back up to the children as shown in the figure below. The playground is 4.70 m above the parking lot, and the school building's vertical wall is h = 6.10 m high, forming a 1.40 m high railing around the playground. The ball is launched at an angle of θ = 53.0∘ above the horizontal at a point d = 24.0 m from the base of the building wall. The ball takes 2.20 s to reach a point vertically above the wall. (Due to the nature of this problem, do not use rounded intermediate values-including answers submitted in WebAssign-in your calculations.) (a) Find the speed (in m/s) at which the ball was launched. m/s (b) Find the vertical distance (in m) by which the ball clears the wall. m (c) Find the horizontal distance (in m) from the wall to the point on the roof where the ball lands. m (d) What If? If the teacher always launches the ball with the speed found in part (a), what is the minimum angle (in degrees above the horizontal) at which he can launch the ball and still clear the playground railing? (Hint: You may need to use the trigonometric identity sec2⁡(θ) = 1+tan2⁡(θ).) ∘ above the horizontal (e) What would be the horizontal distance (in m) from the wall to the point on the roof where the ball lands in this case? m
Solution
0 0

"n" NMOS pass transistors are placed in series as shown, making the delay from point x to y large. (a) Why is the delay from point x to y long? Write an expression for the delay assuming that every NMOS pass transistor has an ON resistance of Ron and a junction capacitance of Ci In order to reduce the delay from point x to y, buffers are inserted along the NMOS pass transistor chain as shown (b) Explain qualitatively, why does inserting buffers cause the delay from point x to y to reduce. (c) Assume that you place a buffer every "m" pass transistors. Also, assume that the delay of each buffer is tbuff. Write an expression for the new delay from point x to y. Explain mathematically why does inserting buffers cause the delay from point x to y to reduce. Finally, explain how you can get the moptimum value which causes to delay from x to y to be minimized. Note: The total number of NNOS' pass transistors is still 'n
Solution
0 0