Virtual Labs

Binomial coefficients and Pascal's triangle

Binomial expansion term of

(a+b)^4+(a-b)^4

for

k=3

will be __

2(^4C_3a^3b)

Explanation

^4C_3a^3b

Explanation

c: 0 Explanation

Explanation

d: none of the above Explanation

Explanation

In this term of binomial expansion of

^6C_2a^2b^4

. which of these is the binomial coefficient?

\frac{6!}{2!}

Explanation

15

Explanation

a^2b^4

Explanation

d: none of these Explanation

Explanation

For what value of

r

does the binomial coefficient

^7C_r

equal

21

5

Explanation

2

Explanation

c: all of the above Explanation

Explanation

d: none of the above Explanation

Explanation

What is

2^n

is equal to?

a: Summation

^nC_r

Explanation

2n

Explanation

n

Explanation

d: none of the above Explanation

Explanation

How many terms are there in the binomial expansion of

(a+b)^6

5

Explanation

7

Explanation

6

Explanation

d: none of the above Explanation

Explanation

If we expand

(x+y)^8

, what is the coefficient of the term

x^5y^3

56

Explanation

28

Explanation

70

Explanation

35

Explanation

In Pascal's Triangle, what is the sum of the squares of the elements in row 4 (starting from row 0)?

a: 70 Explanation

Explanation

b: 36 Explanation

Explanation

c: 100 Explanation

Explanation

d: 82 Explanation

Explanation

What is the value of

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

where n is a positive integer?

2^n

Explanation

-1

Explanation

0

Explanation

1

Explanation

\binom{n}{r} = \binom{n}{r+1}

, what is the value of r in terms of n?

r = \frac{n-1}{2}

Explanation

r = \frac{n}{2}

Explanation

r = n-1

Explanation

r = \frac{n+1}{2}

Explanation

What is the coefficient of

x^3

in the expansion of

(1+x+x^2)^3

a: 7 Explanation

Explanation

b: 10 Explanation

Explanation

c: 6 Explanation

Explanation

d: 12 Explanation

Explanation

\binom{n}{r} \cdot \binom{r}{k} = \binom{n}{k} \cdot \binom{n-k}{r-k}

, this represents which combinatorial identity?

a: Vandermonde's Identity Explanation

Explanation

b: Pascal's Identity Explanation

Explanation

c: Chu-Vandermonde Identity Explanation

Explanation

d: None of the above Explanation

Explanation