Aanchal C.
asked • 06/19/15factorial maths problem help required....
if n is an integer between 0 and 21 ; then the minimum value of n!(21-n)! is:
- 10!11!
- 9!12!
- 21!
- 20!
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2 Answers By Expert Tutors
David W. answered • 06/19/15
Tutor
4.7
(90)
Experienced Prof
This is a good problem to learn about factorials (and to remember that 0! = 1) and to think about multiplying very large numbers by very small numbers.
Extra note: In computers, scientific notation is often helpful in not losing precision when performing many calculations (because computer registers and memory locations are limited in size, like 32-bit and 64-bit), but keeping intermediate calculations within a moderate range (using minimums and maximums) will produce a more accurate final answer.
O.K. 0! = 1 , so the products for n!*(21-n)! start at 21! and get increasingly smaller until they are minimum at 10!*11! and also 11!*10!, and then they get increasingly larger until they get back to 21! [ note: that is 21!*(21-21)! Because n goes from 0 to 21, right? ].
- - - - - - - - - - - - - - - -
Note for Meira: Answer 3 and 4 -- that is, 21! and 20! -- fit the formula as evaluated values for n!(21-n)! when n=0 (allowed) and when n=21 (allowed), but they are maxima for n goes from 0 to 21.
Extra note: In computers, scientific notation is often helpful in not losing precision when performing many calculations (because computer registers and memory locations are limited in size, like 32-bit and 64-bit), but keeping intermediate calculations within a moderate range (using minimums and maximums) will produce a more accurate final answer.
O.K. 0! = 1 , so the products for n!*(21-n)! start at 21! and get increasingly smaller until they are minimum at 10!*11! and also 11!*10!, and then they get increasingly larger until they get back to 21! [ note: that is 21!*(21-21)! Because n goes from 0 to 21, right? ].
- - - - - - - - - - - - - - - -
Note for Meira: Answer 3 and 4 -- that is, 21! and 20! -- fit the formula as evaluated values for n!(21-n)! when n=0 (allowed) and when n=21 (allowed), but they are maxima for n goes from 0 to 21.
Meira P.
Ah ha!!!! Thank you David!!!
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06/19/15
Meira P. answered • 06/19/15
Tutor
5
(1)
30 years teaching Math and Hebrew
n!(21-n)!
10!(21-10)!=10!(11!) = 10*9*8*7*6*5*4*3*2*1*(11*10*9*8*7*6*5*4*3*2*1= do the arithmetic
9!(21-9!)=9!(12!) = set it up as above and do the arithmetic.
The tricky part that Andrew missed is.... are answers 3 & 4 potential answers to the original equation?
It doesn't matter if they're smallest answer listed unless they match the function.
So we make a table:
n n!(21-n)! answer solve as above and answer
1 1!(21-1)! 1!(20!) = 1(20!) = 20! ***ah, so this is in our list!*** (continue to solve as above)
2 2!(21-2)! 2!(19!) = 2(19!) (continue to solve as above)
3 3!(21-3)! 3!(18!) =6(18!) (continue to solve as above)
etc
Can you tell me.... is it possible for answer 3 to be correct? Why or why not?
Andrew M.
I did not miss that. The first thing I looke dat was whether or not all the answers fit the equation as stated. They do and the minimum was
10!(11!). Just FYI.
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06/19/15
Andrew M.
Hiwever, I should have said that. Thank you for pointing out that it needed to be done. :-)
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06/19/15
David W.
"minimum value" means "minimum evaluated value of the function" and may be reduced, simplified, transformed, rounded, truncated, ... before presented as a solution.
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06/19/15
Andrew M.
Actually, whether answer 3 or 4 fit the formula depends on whether or not 0 and 21 are actually included in the allowable values of n, which is debatabl with the way the question is worded. Are 0 and 21 considered to be "between 0 and 21" ... The problem should have said "inclusive" or possibly "from" 0 to 21 instead of between. Regardless, the smallest value was still 10!(11!)
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06/19/15
David W.
Well, just "between" you and me, there is little ambiguity. That is why I think question writers have too much fun wording questions to have a "brick to stumble over" or a brick wall to run into" and look at lots and lots of wrong answers to find the best possible distractors to students (because guessing should not likely produce good grades).
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06/19/15
David W.
Oh, and if 0 is not allowed, the problem can't catch the poor student who doesn't remember that 0!=1 (definition).
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06/19/15
Aanchal C.
firstly, thank you all for the help..but is there any trick or quick method to solve such questions because the format i am working on for these questions requires me to do them in 1 or 2 minutes of maximum time, so long calculating method will take too much of my time ........ please suggest..
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06/19/15
David W.
Aanchal, THX for the excellent question !!
Most people learn by repetition, repetition, repetition (oh, and did I mention repetition?). However, some math can be learned by example or even by precept/concept. That's why when you see "rules" or "magic numbers" or "common tricks" (like giving the edge distance in feet and asking for the answer in square yards), you should have a "tool box" of as many of these tools as you can remember.
Sometimes, teachers and test writers think that being creative enough to solve such problems indicates intelligence (for example, you may see free sample tests at the MENSA genius society web site -- mensa.com). There are also many books and web sites devoted to developing this creativity -- see Luminosity.com). I'm currently in contact with a STEM (science-technology-engineering-math) private high school academy that wants to provide such education to qualified students.
And, sadly, students who could afford books, tutors, review sessions, etc. have scored much higher on the SAT than students who could not afford them. So, the SAT is changing and free Khan Academy tutorials on SAT topics are available on the internet.
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06/19/15
Aanchal C.
Sir,
Thank you again for this helpful piece of advice.... :)
Thank you again for this helpful piece of advice.... :)
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06/19/15
David W.
I’ve looked at some web resources on multiple-choice questions and have found a book and web site that has lots of information to help you:
Multiple Choice Secrets, by Complete Test Preparation
http://www.amazon.com/Multiple-Choice-Secrets-Increase-Score/dp/1477539883/ref=sr_1_1?ie=UTF8&qid=1434732246&sr=8-1&keywords=multiple-choice+tests
http://www.test-preparation.ca/
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06/19/15
Aanchal C.
I have gone through the website and it looks great and helpful.....
searching it for some more tips etc....
thnkx again ..:))
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06/20/15
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Andrew M.
06/19/15