Respuesta :
Answer:
a) 1820
b) 55
c) 10
d) 550
e) 0.3022
Step-by-step explanation:
Data provided in the question:
Number of seniors = 11
Number of juniors = 5
n = number of choices
r = number of person to be selected
number of ways = [tex]^nC_r[/tex]
a) 4-sprinter relay teams can be formed from the group of 16 sprinters
n = 16
r = 4
= ¹⁶C₄
= [tex]\frac{16!}{4!(16-4)!}[/tex]
= [tex]\frac{16\times15\times14\times13\times12!}{4\times3\times2\times1(12!)}[/tex]
=
b) two seniors be chosen to be part of the relay team
n = 11
r = 2
= ¹¹C₂
= [tex]\frac{11!}{2!(11-2)!}[/tex]
= [tex]\frac{11\times10\times9!}{2\times1(9!)}[/tex]
=
c) two juniors be chosen to be part of the relay team
n = 5
r = 2
= ⁵C₂
= [tex]\frac{5!}{2!(5-2)!}[/tex]
= [tex]\frac{5\times4\times3!}{2\times1(3!)}[/tex]
= 10
d) two seniors and two juniors be chosen to be part of the relay team
= two seniors be chosen to be part of the relay team × two juniors be chosen to be part of the relay team
= 55 × 10
= 550
e) P( two seniors and two juniors will be chosen for the relay team )
= [ two seniors and two juniors be chosen to be part of the relay team ] ÷ [4-sprinter relay teams can be formed from the group of 16 sprinters ]
= 550 ÷ 1820
= 0.30219 ≈ 0.3022
By finding the different numbers of combinations using C(N, K), we will get:
- a) C = 1,820
- b) C = 55
- c) C = 3
- d) C = 165
- e) P = 0.091
How to find the number of combinations?
If we have a set of N elements, the number of different sets of K elements that we can make out of these N, is:
[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]
a) We want to select 4 runners out of a group of 16, so we have:
[tex]C(16, 4) = \frac{16!}{(16 - 4)!*4!} = \frac{16*15*14*13}{4*3*2} = 1,820[/tex]
b) We want to select 2 elements out of a group of 11, so we have:
[tex]C(11, 2) = \frac{11!}{(11 - 2)!*2!} = \frac{11*10}{2} = 55[/tex]
c) Now we want to select 2 elements out of a group of 5:
[tex]C(5, 2) = \frac{5!}{(5 - 2)!*2!} = 3[/tex]
d) Here we need to take the product of the two previous numbers of combinations:
C = 55*3 = 165
e) The probability will be given by the quotient between the combinations that have 2 seniors and 2 juniors (the number we got above) and the total number of combinations. This is:
P = 165/1,820 = 0.091
If you want to learn more about probability, you can read:
https://brainly.com/question/251701
