Permutations and Combinations Test 56

Result

Permutations and Combinations Test 56

Score
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Weekly Quiz Competition

Question 1
1 / -0

$$\left( ^ { 2 n } C _ { 0 } \right) ^ { 2 } - \left( ^ { 2 n } C _ { 1 } \right) ^ { 2 } + \left( ^ { 2 n } C _ { 2 } \right) ^ { 2 } - \ldots + ( - 1 ) ^ { 2 n } \left( ^ { 2 n } C _ { 2 n } \right) ^ { 2 }$$ equals to

A
$$\left( ^ { 2 n } C _ { n } \right) ^ { 2 }$$

B
$$( - 1 ) ^ { 2 n } \left( ^ { 2 n } C _ { n } \right)$$

C
$$( - 1 ) ^ { n } \left( ^ { 2 n } C _ { n } \right)$$

D
$$\left( ^ { n } C _ { n } \right) + \left( ^ { 2 n } C _ { n } \right)$$

Solution
Question 2
1 / -0

If $${C_r} = {\,^n}{C_r}\,\,and\,\,\left( {{C_0} + {C_1}} \right)\,\,\left( {{C_1} + {C_2}} \right)......$$
$$\left( {{C_{n - 1}} + {C_n}} \right) = k\frac{{{{\left( {n + 1} \right)}^n}}}{{n!}}$$ then the value of k is

A
$${C_0}.{C_1}.{C_2}....{C_n}$$

B
$${\left( {{C_1}} \right)^2}.{\left( {{C_2}} \right)^2}....{\left( {{C_n}} \right)^2}$$

C
$${C_1} + {C_2} + .... + {C_n}$$

D
None of these

Solution
Question 3
1 / -0

If $$^{35}C_{n+7}=^{35}C_{4n-2}$$, then the value of $$n$$ is

A
$$3$$

B
$$4$$

C
$$5$$

D
$$6$$

Solution

Given
$$^{35}C_{n+7}=^{35}C_{4n-2}$$
It is in form if
$$^{n}C_{x}=^{n}C_{v}$$, then
x=y or x+y=n
now
(i)
n+7=4n-2
4n-n=7+2
3n=9
n=3
(ii)
n+7+4n-2=35
5n+5=35
5n=30
n=6
Question 4
1 / -0

In a conference hall 10 executive are to be seated on 10 chairs placed left to right. Executive $$E_1$$ wants to sit to the right of $$E_2$$ Also, $$E_4$$ wants to sit the left of $$E_1$$ and $$E_5$$ wants to sits to the left of ways $$E_4$$ Number of ways the executives can be seated on the 10 chairs, is

A
$$\dfrac{10!}{4!}$$

B
$$\dfrac{10!}{3}$$

C
$$10!$$

D
$$3 \times \dfrac{10!}{4!}$$
Question 5
1 / -0

Total number of four-digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are

A
216

B
375

C
400

D
720
Question 6
1 / -0

What is the hundreds digits of ( 20 !-15!) ?

A
0

B
1

C
2

D
4

E
5

Solution
Question 7
1 / -0

If $$n$$ and $$r$$ are positive integers such that $$r < n$$ then $$^{n}C_{r}+^{n}C_{r-1}$$ is equal to

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

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

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

D
$$2^{n}C_{r}$$

Solution
Question 8
1 / -0

The number of five digit numbers in which digits decrease from left to right, is

A
$$^{9}C_{5}$$

B
$$^{10}C_{5}$$

C
$$2\times ^{10}C_{5}$$

D
$$^{10}P_{5}$$
Question 9
1 / -0

If $$x\epsilon N and C^{x-1_{4}}-C^{X-1}_{3}-\frac{5}{4}P^{x-2}_{2}< 0$$ then x is equal to

A
4,5,6,7,8,9,10

B
3

C
$$3\leq x\leq 10$$

D
none of these

Solution
Question 10
1 / -0

If $$m$$ and $$n$$ are positive integers more than or equal to $$2 , m > n ,$$ then $$( m n ) !$$ is divisible by

A
$$( m ! ) ^ { n }$$

B
$$( n ! ) ^ { m }$$

C
$$( m + n ) !$$

D
$$( m - n ) !$$

Solution