Lesson test: Arrangements and selections

Arrangements and selections

1.[1p]
Twelve finalists race for gold, silver and bronze. In how many ways can the three medals be awarded?

CorrectNot quite: 1320

2.[1p]
How many committees of $4$ can be chosen from $12$ people?

CorrectNot quite: 495

3.[2p]
How many distinct arrangements does the word LETTERS have?

CorrectNot quite: 1260

4.[2p]
Four identical dice are rolled together. How many distinguishable outcomes are there?

CorrectNot quite: 126

5.[2p]
How many solutions does $x_{1} + x_{2} + x_{3} = 10$ have in nonnegative integers?

CorrectNot quite: 66

6.[2p]
Twelve identical coins are shared among four children so that every child receives at least one. In how many ways can this be done?

CorrectNot quite: 165

7.[3p]
How many lattice paths of unit steps right and up run from $(0, 0)$ to $(5, 5)$ without passing through $(2, 2)$ ?

CorrectNot quite: 132

8.[2p]
Eight different books are handed out to three students, and a student may receive none. Which count is correct?

$3^{8} = 6561$ , since each book independently chooses its recipient

Correct

$(\binom{10}{2}) = 45$ , by stars and bars with $8$ stars and $2$ bars

The answer is: $3^8 = 6561$, since each book independently chooses its recipient

$P (8, 3) = 336$ , since three books are placed in order

The answer is: $3^8 = 6561$, since each book independently chooses its recipient

$(\binom{8}{3}) = 56$ , since three of the eight books are chosen

The answer is: $3^8 = 6561$, since each book independently chooses its recipient

9.[3p]
Match each kind of selection of $k$ things from $n$ kinds to its count.

Ordered, repetition allowed
Ordered, no repetition
Unordered, no repetition
Unordered, repetition allowed

$(\binom{n}{k})$
$(\binom{n + k - 1}{k})$
$\frac{n!}{(n - k)!}$
$n^{k}$

Show the answer

Ordered, repetition allowed: $n^{k}$ Ordered, no repetition: $\frac{n!}{(n - k)!}$ Unordered, no repetition: $(\binom{n}{k})$ Unordered, repetition allowed: $(\binom{n + k - 1}{k})$

1.[1p] Twelve finalists race for gold, silver and bronze. In how many ways can the three medals be awarded?

2.[1p] How many committees of 4 can be chosen from 12 people?

3.[2p] How many distinct arrangements does the word LETTERS have?

4.[2p] Four identical dice are rolled together. How many distinguishable outcomes are there?

5.[2p] How many solutions does x1+x2+x3=10 have in nonnegative integers?

6.[2p] Twelve identical coins are shared among four children so that every child receives at least one. In how many ways can this be done?

7.[3p] How many lattice paths of unit steps right and up run from (0,0) to (5,5) without passing through (2,2)?

8.[2p] Eight different books are handed out to three students, and a student may receive none. Which count is correct?

9.[3p] Match each kind of selection of k things from n kinds to its count.

1.[1p]
Twelve finalists race for gold, silver and bronze. In how many ways can the three medals be awarded?

2.[1p]
How many committees of $4$ can be chosen from $12$ people?

3.[2p]
How many distinct arrangements does the word LETTERS have?

4.[2p]
Four identical dice are rolled together. How many distinguishable outcomes are there?

5.[2p]
How many solutions does $x_{1} + x_{2} + x_{3} = 10$ have in nonnegative integers?

6.[2p]
Twelve identical coins are shared among four children so that every child receives at least one. In how many ways can this be done?

7.[3p]
How many lattice paths of unit steps right and up run from $(0, 0)$ to $(5, 5)$ without passing through $(2, 2)$ ?

8.[2p]
Eight different books are handed out to three students, and a student may receive none. Which count is correct?

9.[3p]
Match each kind of selection of $k$ things from $n$ kinds to its count.