Virtual Labs

Binomial coefficients and Pascal's triangle

For what value of

k

is the binomial coefficient of

a^{k}b^{5-k}

in the expansion of

(a+b)^5

equals

10

2

Explanation

3

Explanation

c: both

a

and

b

Explanation

d: none of the above Explanation

Explanation

Which of these constructions define Pascal's triangle?

a: binomial expansion of

(a+b)^c

where

c

is a constant and

a

b<=i

where

i

is the row number Explanation

Explanation

b: binomial coefficient

^{n}C_{r}

where

n

is the row number and

r

is the column number both starting at

0

Explanation

c: binomial expansion of

c^i

where

c

is a constant and

i

is the row number Explanation

Explanation

d: none of the above Explanation

Explanation

What does the summation of a row of Pascal's triangle represent?

a: binomial expansion of

(a+b)^n

for

a=b=1

where

n

is the number of elements in the row Explanation

Explanation

b: binomial expansion of

(a+b)^n

for

a=b=1

where

n

is the row index starting from

0

at the top of the triangle Explanation

Explanation

c: square of the summation of elements of the previous row Explanation

Explanation

d: none of the above Explanation

Explanation

If we add the values of second column third row in pascal's triangle for

(a+b)^n

and

(b+c)^n

, we get ______

ab+bc

Explanation

2(ab+bc)

Explanation

4

Explanation

d: none of these Explanation

Explanation

How can an element of Pascal's triangle be derived using existing elements?

a: by adding all the elements in the previous row and multiplying by the row index Explanation

Explanation

b: by adding the two elements directly on top of it in the previous row Explanation

Explanation

c: by adding all the elements in the previous row and multiplying by

2

Explanation

d: none of the above Explanation

Explanation

In Pascal's triangle, what is the relationship between

^nC_r

and

^nC_{n-r}

a: They are consecutive numbers Explanation

Explanation

b: They are equal Explanation

Explanation

c: Their sum equals

2^n

Explanation

d: Their product equals

^nC_n

Explanation

What is the sum of squares of binomial coefficients in the expansion of

(x+y)^4

a: 70 Explanation

Explanation

b: 35 Explanation

Explanation

c: 14 Explanation

Explanation

d: 28 Explanation

Explanation

^{10}C_r = ^{10}C_{r+1}

, what is the value of

r

a: 4 Explanation

Explanation

b: 5 Explanation

Explanation

c: 6 Explanation

Explanation

d: 7 Explanation

Explanation

In Pascal's triangle, if you multiply each number in row 5 by its position number (starting from 0), what is the sum of these products?

a: 80 Explanation

Explanation

b: 160 Explanation

Explanation

c: 120 Explanation

Explanation

d: 100 Explanation

Explanation

P(n,r)

represents the sum of all products formed by taking

r

different numbers from row

n

of Pascal's triangle, what is

P(4,2)

a: 70 Explanation

Explanation

b: 85 Explanation

Explanation

c: 100 Explanation

Explanation

d: 110 Explanation

Explanation

In Pascal's triangle, what is the value of

\sum_{k=0}^n k^2\binom{n}{k}

in terms of

n

n2^{n-1}

Explanation

n(n+1)2^{n-2}

Explanation

n^22^{n-2}

Explanation

n(n+1)2^{n-1}

Explanation