Lesson test: Binomial coefficients

Binomial coefficients

1.[2p]
Find the coefficient of $x^{3}$ in $(2 x + 1)^{6}$ .

CorrectNot quite: 160

2.[2p]
Find the coefficient of $x^{2} y^{3}$ in $(x - 2 y)^{5}$ .

CorrectNot quite: -80

3.[1p]
What is $(\binom{10}{0}) + (\binom{10}{1}) + (\binom{10}{2}) + \dots + (\binom{10}{10})$ ?

CorrectNot quite: 1024

4.[2p]
Compute ${(\binom{6}{0})}^{2} + {(\binom{6}{1})}^{2} + \dots + {(\binom{6}{6})}^{2}$ .

CorrectNot quite: 924

5.[3p]
A panel of four is chosen from five chemists and six biologists. How many panels include at least two chemists?

CorrectNot quite: 215

6.[2p]
Compute $(\binom{2}{2}) + (\binom{3}{2}) + (\binom{4}{2}) + \dots + (\binom{9}{2})$ .

CorrectNot quite: 120

7.[2p]
Which double count proves $k (\binom{n}{k}) = n (\binom{n - 1}{k - 1})$ ?

Committees of $k$ from $n$ people with one member marked as chair, counted committee first and then chair first

Correct

Committees of $k$ from $n$ people, split by whether one fixed person is on them

The answer is: Committees of $k$ from $n$ people with one member marked as chair, counted committee first and then chair first

Subsets of an $n$ element set, sorted by their size

The answer is: Committees of $k$ from $n$ people with one member marked as chair, counted committee first and then chair first

Subsets of size $k$ , sorted by their largest element

The answer is: Committees of $k$ from $n$ people with one member marked as chair, counted committee first and then chair first

8.[3p]
Match each identity to the way its set is split in the counting proof.

Pascal's rule
Sum of a row
Vandermonde's identity
Hockey stick

committees split by how many members come from the first group
subsets split by their size
subsets split by their largest element
committees split by whether one fixed person is on them

Show the answer

Pascal's rule: committees split by whether one fixed person is on them Sum of a row: subsets split by their size Vandermonde's identity: committees split by how many members come from the first group Hockey stick: subsets split by their largest element

9.[3p]
Which of these statements are correct?

Select all that apply

A nonempty finite set has as many subsets of even size as of odd size

Correct

$(\binom{n}{k}) = (\binom{n}{n - k})$ , because sending a subset to its complement is a bijection

Correct

$\sum_{k = 0}^{n} k (\binom{n}{k}) = n \cdot 2^{n - 1}$

Correct

$(\binom{n}{k}) = (\binom{n - 1}{k - 1}) \cdot (\binom{n - 1}{k})$

The answer is: A nonempty finite set has as many subsets of even size as of odd size, $(\binom{n}{k}) = (\binom{n}{n - k})$ , because sending a subset to its complement is a bijection, $\sum_{k = 0}^{n} k (\binom{n}{k}) = n \cdot 2^{n - 1}$

The coefficient of $x^{n - k} y^{k}$ in $(x + y)^{n}$ is $(\binom{n}{k})$ only when $n$ is even

1.[2p] Find the coefficient of x3 in (2x+1)6.

2.[2p] Find the coefficient of x2y3 in (x−2y)5.

3.[1p] What is (100)+(101)+(102)+⋯+(1010)?

4.[2p] Compute (60)2+(61)2+⋯+(66)2.

5.[3p] A panel of four is chosen from five chemists and six biologists. How many panels include at least two chemists?

6.[2p] Compute (22)+(32)+(42)+⋯+(92).

7.[2p] Which double count proves k(nk)=n(n−1k−1)?

8.[3p] Match each identity to the way its set is split in the counting proof.

9.[3p] Which of these statements are correct?

1.[2p]
Find the coefficient of $x^{3}$ in $(2 x + 1)^{6}$ .

2.[2p]
Find the coefficient of $x^{2} y^{3}$ in $(x - 2 y)^{5}$ .

3.[1p]
What is $(\binom{10}{0}) + (\binom{10}{1}) + (\binom{10}{2}) + \dots + (\binom{10}{10})$ ?

4.[2p]
Compute ${(\binom{6}{0})}^{2} + {(\binom{6}{1})}^{2} + \dots + {(\binom{6}{6})}^{2}$ .

5.[3p]
A panel of four is chosen from five chemists and six biologists. How many panels include at least two chemists?

6.[2p]
Compute $(\binom{2}{2}) + (\binom{3}{2}) + (\binom{4}{2}) + \dots + (\binom{9}{2})$ .

7.[2p]
Which double count proves $k (\binom{n}{k}) = n (\binom{n - 1}{k - 1})$ ?

8.[3p]
Match each identity to the way its set is split in the counting proof.

9.[3p]
Which of these statements are correct?