1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
dexar [7]
2 years ago
9

Let k be a positive integer. In how many ways can one select three distinct numbers from the set {1,2,..., 3k} such that their s

um is divisible by 3
Mathematics
1 answer:
diamong [38]2 years ago
8 0

Reduce the numbers in the list modulo 3 to get the set

{1, 2, 0, 1, 2, 0, …, 1, 2, 0}

containing k copies each of 1, 2, and 0.

Take any 3 elements from the list. Their sum is divisible by 3 if those elements' residues also sum to 3 ≡ 0 (mod 3). To get a sum of 0, we must make one of the following choices:

  • 3 elements each with the same residue, so

0 + 0 + 0 ≡ 0 (mod 3)

1 + 1 + 1 ≡ 3 ≡ 0 (mod 3)

2 + 2 + 2 ≡ 6 ≡ 0 (mod 3)

  • 1 element each with different residues, so

0 + 1 + 2 ≡ 3 ≡ 0 (mod 3)

There are

\dbinom k3 \dbinom k0 \dbinom k0 = \dfrac{k(k-1)(k-2)}6

ways of choosing 3 elements with a given residue and 0 elements with any other residue, hence

3\dbinom k3\dbinom k0\dbinom k0 = \dfrac{k(k-1)(k-2)}2

ways of choosing any 3 elements with the same residue, and there are

\dbinom k1 \dbinom k1 \dbinom k1 = k^3

ways of choosing any 3 elements with distinct residues.

So, the total number of ways of making the selection is

3\dbinom k3\dbinom k0^2 + \dbinom k1^3 = \boxed{\dfrac32 k^3 - \dfrac32 k^2 - k}

You might be interested in
June’s portfolio includes 177 shares of parchar special effects, 213 shares of giaco household goods, and 152 shares of raxin ac
givi [52]
Junes is it a joke or not even a cheer leader for the last year of her and I can’t bc
5 0
1 year ago
Find the product of 2.7 and 3.59.
Paha777 [63]

Answer:

9.693 hope this helps!!!

3 0
2 years ago
Use the net to find the surface area of the regular pyramid.<br><br> plz, help I'm stuck.
Zepler [3.9K]

Answer:

178.3 mm²

Step-by-step explanation:

Surface area = Base area + 3 x side triangles area

Sₐ = 43.3 + 3 x (1/2 x (10 x 9)) = 43.3 + (3 x 45) = 178.3

5 0
3 years ago
Which numbers are​ irrational?
Zepler [3.9K]

Answer:

54  hope this helps

Step-by-step explanation:

5 0
3 years ago
At a school fair, students were challenged to hit one of the small congruent circles on the large rectangular board with a ball.
shtirl [24]

Answer:

~4.7%

Step-by-step explanation:

Area of the rectangle: 35*52=1820

Area of 1 small circle: A=πr2=π·32≈28.27433

28.27433*3=84.82299

84.82299/1820=0.04660603846

Therefore, about 4.7%

6 0
2 years ago
Other questions:
  • Which of the following are solutions to the equation below (5x+1)^2=7?<br><br> Check all that apply.
    14·2 answers
  • Large reusable bottles cost $4 more than small ones. Eight large bottles cost $24 less than 12 small ones. How much does one sma
    13·1 answer
  • X^3 = 125 <br><br> x = ? <br><br> Help!!!!
    8·2 answers
  • Paul needs to buy paint to cover all outside surfaces of his shed, depicted below. How much area, in square yards, does he need
    9·1 answer
  • Which has the greater energy, light of wavelength 519 nm or light with a frequency of 5.42 x 10^8 sec^-1?
    11·1 answer
  • Identify the correct equation that describe the relationship between a sine and cosine wave. a. v(t) = A sin (2πft + π/2) = A co
    12·1 answer
  • Help me real quickhahahahaha
    11·2 answers
  • Given the function g(x)=x3−x,<br> what is the value of g(3)?
    13·2 answers
  • What are the solutions to this quadratic equation? 5x^2-7-18=0
    15·1 answer
  • The value of y varies directly with x if x=8 then y=96 what is the value of x when y=196
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!