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A space program is recruiting a team of astronauts to journe

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Senior Manager
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A space program is recruiting a team of astronauts to journe [#permalink]  04 Oct 2017, 20:36
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22% (02:22) correct 77% (03:02) wrong based on 9 sessions

A space program is recruiting a team of astronauts to journey to Mars aboard a four-person shuttle. If the number of possible teams is less than 100 but greater than 10, then what is the possible number of astronauts who did not make the space team? Indicate all such numbers.

[A] 9

[B] 8

[C] 7

[D] 6

[E] 5

[F] 4

[G] 3

[H] 2

[I] 1

[Reveal] Spoiler: OA
F, G, H

Kudos for correct solution.
[Reveal] Spoiler: OA
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Re: A space program is recruiting a team of astronauts to journe [#permalink]  04 Oct 2017, 23:00
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Bunuel wrote:

A space program is recruiting a team of astronauts to journey to Mars aboard a four-person shuttle. If the number of possible teams is less than 100 but greater than 10, then what is the possible number of astronauts who did not make the space team? Indicate all such numbers.

[A] 9

[B] 8

[C] 7

[D] 6

[E] 5

[F] 4

[G] 3

[H] 2

[I] 1

[Reveal] Spoiler: OA
F, G, H

Kudos for correct solution.

Now here the total number of astronount choosen is not given, let it be = x

now we have to choose 4 astronount from the x astronounts

i.e, we can re write as xC4 and that has to be between 10 and 100

Now if x= 5 then the value becomes = 5 (5C4)

if we choose x=6 then the value becomes = 15 (6C4)

if we choose x= 8 then the value becomes =70 (8C4)

if we choose x= 9 then value becomes = 126 (9C4)

therefore it should lie between 6 and 8 .

Now we need the astronauts who did not meet i.e we have to subtract 4 from the total numbers

i.e 6-4 =2 and 8-4 = 4 (minimum and maximum values)

Therefore numbers 2 ,3, 4 did not made it.

Kindly let me know if my reasoning is correct
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Manager
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Re: A space program is recruiting a team of astronauts to journe [#permalink]  09 Jan 2019, 23:52
Expert's post
Lets use very simple algebra here.

the total number of combinations (N) is less than 100 but greater than 10
the total team members is 4
thus 10< Nc4 <100

10 < Nc4 >100

10 < N!/((4!)*N-4)!) < 100

multiply both sides by 4!

10*4! < N!/(N-4)! <4!*100

240 < N!/(N-4)! <2400

240 < N(N-1)(N-2)(N-3) < 2400

Now, keep in mind they are asking for how many people out of the group was not chosen, so N = 4+ the answer choices given

since we know that

240 < N(N-1)(N-2)(N-3) < 2400
then the 4 multiplied numbers cant all be greater than 10 because we would get a number greater than 10000.

240 < 10(9)(8)(7)=5040, which is not less than 2400
try N=9

240 < (9)(8)(7)(6)= 3024, which is not less than 2400

Try N=8
240 < (8)(7)(6)(5)= 1680< 2400, then 8-4 = 4 people not taken

Try N=7

240 < (7)(6)(5)(4)=840 <2400 7-4 = 3 people not taken

Try N= 6

240 < (6)(5)(4)(3) < 360 6-4 = 2 people not taken

At N = 5

5*4*3*2= 120 <240 which cant work

So the number of people left behind are 4,3,2
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Re: A space program is recruiting a team of astronauts to journe [#permalink]  10 Jan 2019, 03:53
Expert's post
Bunuel wrote:

A space program is recruiting a team of astronauts to journey to Mars aboard a four-person shuttle. If the number of possible teams is less than 100 but greater than 10, then what is the possible number of astronauts who did not make the space team? Indicate all such numbers.

[A] 9

8

[C] 7

[D] 6

[E] 5

[F] 4

[G] 3

[H] 2

[I] 1

[Reveal] Spoiler: OA
F, G, H

Kudos for correct solution.

[b]So, the question is about possible number of teams, thus a combination problem--- nCr
In nCr, r is 4, so nC4..

What is n? - It is given in the choices, that is CHOICE + 4
so (9+4)C4, (8+4)C4, (7+4)C4,....(2+4)C4, (1+4)C4..

Also $$10<nC4<100$$...
1) $$nC4>10$$
Take n as the least (1+4) so 5C4=5, which is less than 10, so take next higher
take n as the 6,so $$6C4 = \frac{6*5}{2}=15$$, greater than 10.. so possible
2)$$nC4<100$$
take n as 13, $$13C4 = \frac{13*12*11*10}{4*3*2}=715$$, way more than 100 so take the middle value in the choice
n =9, so $$9C4 = \frac{9*8*7*6}{4*3*2}=126$$, so >100
n=8, so $$8C4 = \frac{8*7*6*5}{4*3*2}=70$$

so n is from 6 to 8
if 4 are sent, the number of astronaut left can be (6-4), (7-4) or (8-4), thus 2, 3, or 4
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: A space program is recruiting a team of astronauts to journe   [#permalink] 10 Jan 2019, 03:53
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