2000002501
Level:
B
Simplify: \( \frac{8!}{6!} \)
\( 56\)
\( 336 \)
\( 8 \)
\(\frac{4}{3} \)
Combinatorics
Word Problems I
Submitted by ladislav.foltyn on Wed, 04/03/2019 - 20:53Combinatorial Equations and Inequalities
Submitted by ladislav.foltyn on Fri, 03/01/2019 - 12:38Question:
\small For each of the given equations and inequalities, where $x$ is a positive integer, choose the correct solution set. (Don’t use a calculator.)
Combinatorial expressions
Question:
Without calculator, simplify and evaluate the following expressions (for $n\in\mathbb N$). For each expression choose its correct value and if it isn’t in the table, choose X.
1003024704
Level:
C
Suppose the coefficient of the quadratic term (i.e. the coefficient at \( x^2 \)) in the expansion of \( (1+x)^n \) equals \( 300 \). Find the value of the exponent \( n \).
\( 25 \)
\( 24 \)
\( 30 \)
\( 20 \)
1003024703
Level:
C
Let the term with \( x^{-3} \) in the expansion of \( \left(4x^{-\frac12}-\frac12\right)^{10} \) equals \( 105 \). Find the value of \( x \).
\( 8 \)
\( 9 \)
\( 10 \)
\( 11 \)
1003024702
Level:
C
Find the constant term in the expansion of \( \left(2+3x^2\right)^{30} \).
\( 2^{30} \)
\( 2^{29} \)
\( 3\cdot2^{30} \)
\( 3\cdot2^{29} \)
1003024701
Level:
C
What is the coefficient at \( x^7 \) in the expansion of \( (1-3x)^9 \)?
\( -78\:732 \)
\( 59\:049 \)
\( -59\:049 \)
\( 78\:732 \)
1003024611
Level:
A
On a lock of a safe deposit box a ten-digit code can be set. The code can consist only of four \( 1 \)s, three \( 2 \)s, two \( 3 \)s, and one \( 4 \). How many ways are there to set the code?
\( \frac{10!}{4!\cdot3!\cdot2!} = 12\:600 \)
\( \frac{10!}{4!+3!+2!}=113\:400 \)
\( 10!-4!\cdot3!\cdot5!=3\:628\:512 \)
\( 10! = 3\:628\:800 \)
1003024610
Level:
A
In a high-speed train set, there should be the following cars included: \( 3 \) first class cars, \( 5 \) second class cars, \( 2 \) sleeping cars, \( 1 \) dining car, and \( 2 \) luggage cars. How many ways are there to arrange the cars in this high-speed train set?
\( \frac{13!}{(2!)^2\cdot3!\cdot5!}=2\:162\:160 \)
\( \frac{13!}{(2!)^2+3!+5!}=47\:900\:160 \)
\( 13!-(2!)^2\cdot3!\cdot5!=6\:227\:017\:920 \)
\( 13!-\left|(2!)^2+3!+5!\right|=6\:227\:020\:670 \)