Permutations and Combinations Test 53

Result

Permutations and Combinations Test 53

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Weekly Quiz Competition

Question 1
1 / -0

2. $$^n$$C$$_0$$ + 2$$^2$$ . $$\dfrac{^nC_1}{2}$$ + 2$$^3$$ . $$\dfrac{^nC_2}{3}$$ +.......+ 2$${n + 1}$$ . $$\dfrac{^nC_n}{n + 1}$$ =

A
3$$^{n+ 1} $$ - 1

B
$$\dfrac{3^{n+1} - 1}{n + 1}$$

C
$$\dfrac{3^{n +1}}{n + 1}$$

D
$$\dfrac{3^{n +1 }-1}{n}$$

Solution
Question 2
1 / -0

If $$(1+x)^{n}=C_{0}+C_{1}x+C_{2}x^{2}+...+C_{n}x^{n}$$, then the value of $$C_{0}^{2}+\dfrac{C_{1}^{2}}{2}+\dfrac{C_{2}^{2}}{2}+.........+\dfrac{C_{n}^{2}}{n+1}$$ is

A
$$\dfrac{(2n-1)!}{\left\{(n+1)!\right\}^{2}}$$

B
$$\dfrac{(2n-1)!}{(n+1)!}$$

C
$$\dfrac{(2n+1)!}{\left\{(n+1)!\right\}^{2}}$$

D
$$\dfrac{(2n)!}{(n+1)!}$$

Solution
Question 3
1 / -0

The sum $$\sum _{ r=1 }^{ n }{ r.{ }^{ 2n }{ C }_{ r } } $$ is equal to.

A
$$n.2^{2n-1}$$

B
$$2^{2n-1}$$

C
$$2^{n-1}+1$$

D
none of these

Solution
Question 4
1 / -0

The number of ways in which three girls and ten boys can be seated in two vans, each having numbered seats, three in the front and four at the back is

A
$${ ^{ 14 }C }_{ 12 }$$

B
$${ ^{ 14 }P }_{ 13 }$$

C
$$14!$$

D
$${ 10 }^{ 3 }$$

Solution
Question 5
1 / -0

The total no. of six digit numbers $${ x }_{ 1 }{ x }_{ 2 }{ x }_{ 3 }{ x }_{ 4 }{ x }_{ 5 }{ x }_{ 6 }$$ having the property $${ x }_{ 1 }<{ x }_{ 2 }\le { x }_{ 3 }<{ x }_{ 4 }<{ x }_{ 5 }\le { x }_{ 6 }$$, is equal to

A
$$_{ }^{ 11 }{ C }_{ 6 }$$

B
$$_{ }^{ 16 }{ C }_{ 2 }$$

C
$$_{ }^{ 17 }{ C }_{ 2 }$$

D
$$_{ }^{ 18 }{ C }_{ 2 }$$

Solution
Question 6
1 / -0

If $$^nC_r$$ denotes the number of combinations of n things taken r things at a time, then the expression $$^nC_{r+1} + ^nC_{r-1}+2 ^nC_r is$$

A
$$^{n+2}C_{r+1}$$

B
$$^{n+1}C_r$$

C
$$^{n+1}C_{r+1}$$

D
$$^{n+2}C_r$$

Solution
Question 7
1 / -0

If $$^{ n }{ C }_{ 3 }=^{ n }{ C }_{ 2 }$$, then n is equal to

A
2

B
3

C
5

D
none of these
Question 8
1 / -0

$$If\,a\, = {\,^m}{C_2},\,\,then{\,^a}{C_{2\,}}\,is\,equal\,to$$

A
$$^{m + 1}{C_4}$$

B
$$^{m - 1}{C_4}$$

C
$$3{ \cdot ^{m + 2}}{C_4}$$

D
$$3{ \cdot ^{m + 1}}{C_4}$$

Solution
Question 9
1 / -0

$$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +\dfrac { 2{ C }_{ 2 } }{ { C }_{ 1 } } +...\dfrac { 3{ C }_{ 3 } }{ { C }_{ n-1 } } =$$

A
$$\dfrac { n\left( n-1 \right) }{ 2 } $$

B
$$\dfrac { n\left( n+2\right) }{ 2 } $$

C
$$\dfrac { n\left( n+1 \right) }{ 2 } $$

D
$$\dfrac { \left( n-1 \right) \left( n-2 \right) }{ 2 } $$

Solution
Question 10
1 / -0

If $$\sum\limits_{r = 0}^n {\left( {\frac{{r + 2}}{{r + 1}}} \right)} \,{\,^n}{C_r} = \frac{{{2^8} - 1}}{6},$$ then 'n' is

A
8

B
4

C
6

D
5

Solution