If $${}^n{C_r}$$ denotes the number of combination of n things taken r at a time, then the expression $$\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \times \,{}^n{C_r}$$ equals
The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by
A
$$7!\, \times 5!\,\,$$
B
$$6!\, \times 5!$$
C
$$30!$$
D
$$5!\, \times 4!$$
Explanation
6 men can sit at the round table = $$\left( {6 - 1} \right)! = 5!$$ ways
Now at the round table among 6 men there are 6 empty places and 5 women can sit at those 6 empty positions.
So total no of ways 6 men and 5 women can dine at the round table
= $$5!\, \times {}^6{C_5} \times 5!$$
= $$5!\, \times 6 \times 5!$$
= $$5!\, \times 6!$$
3
AIEEE 2003
MCQ (Single Correct Answer)
A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
A
346
B
140
C
196
D
280
Explanation
Case 1 :
No of ways student can answer 10 questions = $${}^5{C_4} \times {}^8{C_6}$$ = 140
Case 2 :
No of ways student can answer 10 questions = $${}^5{C_5} \times {}^8{C_5}$$ = 56
$$\therefore$$ Total ways = 140 + 56 = 196
4
AIEEE 2002
MCQ (Single Correct Answer)
The sum of integers from 1 to 100 that are divisible by 2 or 5 is
A
3000
B
3050
C
3600
D
3250
Explanation
According to this question, any number between 1 to 100 should be divisible by 2 or 5 but not by 2$$ \times $$5 = 10.
Possible numbers between 1 to 100 divisible by 2 are 2, 4, 6, .... , 100
This is an A.P where first term = 2, last term = 100 and total terms = 50.
$$ \therefore $$ Sum of the numbers divisible by 2
= $${{50} \over 2}\left[ {2 + 100} \right]$$
= 25$$ \times $$102
= 2550
Possible numbers between 1 to 100 divisible by 5 are 5, 10, 15, .... , 100
$$ \therefore $$ Sum of the numbers divisible by 5
= $${{20} \over 2}\left[ {5 + 100} \right]$$
= 10$$ \times $$105
= 1050
And possible numbers between 1 to 100 divisible by 10 are 10, 20, 30, .... , 100
$$ \therefore $$ Sum of the numbers divisible by 10