Discrete Mathematics - Test Questions II

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Question 11

Question:   Out of 10 consonants and 4 vowels, how many words can be formed each containing 6 constants and 3 vowels?

Answer:    6 consonants can be chosen out of 10 in 10C6 ways and 3 vowels can be drawn out of 4 in 4C3 ways.
Total number of selection of 6 consonants and 3 vowels is

Question 12

Question:   There are n concurrent lines and another line parallel to one of them. How many different triangles can be formed by (n+1) lines?

Answer:    Number of triangles = number of selection of 2 lines from the (n-1) lines which are cut by the last line.

Question 13

Question:   A man has 7 relatives, 4 of them are ladies and 3 gentlemen. His wife has 7 relatives, 4 of them gentlemen and 3 ladies. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen, so that there are 3 of the man's relatives and 3 of the wife's relatives?

Answer:   
We can express all possibilities in the following table:


\The total number of ways = 16 + 324 + 144 + 1 = 485

Question 14

Question:   How many four digit numbers can be formed using the digits 3,4,6,7 and 8 without repetition?

Answer:   
Unit's place can be filled in 5 ways (First Digit).
Ten's place can be filled in 4 ways (Second Digit).
Hundred's place can be filled in 3 ways (Third Digit).
Thousand's place can be filled in 2 ways (Fourth digit).
\By fundamental principle of counting, the number of four digit numbers are 2 x 3 x 4 x 5 = 120.

Question 15

Question:   In a railway compartment, there are 6 seats available on a bench. How many ways can 4 passengers occupy the six seats?

Answer:    The first passenger can occupy any of the 6 vacant seats. The second passenger can occupy any of the 5 remaining seats. The third passenger can occupy any of the 4 remaining seats and the fourth passenger can occupy any of the 3 remaining seats.
By the fundamental principle of counting, the total number of ways is 6 x 5 x 4 x 3 = 360.

Question 16

Question:  

Answer:   
We know that the (r + 1)th term of (x + a)n is given by









Question 17

Question:  

Answer:    Expanding by binomial theorem, we get






Question 18

Question:  

Answer:   









Question 19

Question:  

Answer:   
To find the two middle terms:

Since n = 9, the two middle terms are



Question 20

Question:  

Answer: