In how many ways a committee, consisting of 5 men and 6 women can be formed from 8 men and 10 women?
Answer: C
The men and the women are chosen from separate pools, so the counts multiply.
\[ {}^{8}C_{5} \times {}^{10}C_{6} \]
\[ = 56 \times 210 = 11{,}760 \]
Using \( {}^{8}C_{3} \) and \( {}^{10}C_{4} \) gives the same values, since choosing whom to include is equivalent to choosing whom to leave out.
Please log in or register to report this question.
How many three letter words are formed using the letters of the word TIME?
Answer: D
TIME has four distinct letters, and order matters in a word.
\[ {}^{4}P_{3} = 4 \times 3 \times 2 = 24 \]
The corresponding combination \( {}^{4}C_{3} = 4 \) would count only which three letters are used, not how they are arranged.
Please log in or register to report this question.
Using all the letters of the word "NOKIA", how many words can be formed, which begin with N and end with A?
Answer: B
The first and last letters are fixed as N and A, leaving O, K and I for the three middle positions.
\[ 3! = 6 \]
Without the restriction all five letters could move, giving \( 5! = 120 \).
Please log in or register to report this question.
Three dice, each having six faces numbered 1 to 6, are rolled. What is the number of possible outcomes in which at least one die shows the number 2?
Answer: C
"At least one" is easiest through the complement.
\[ \text{total outcomes} = 6^{3} = 216 \]
\[ \text{no die shows 2} = 5^{3} = 125 \]
\[ 216 - 125 = 91 \]
Counting exactly one, exactly two and exactly three 2's gives \( 75 + 15 + 1 = 91 \) as well, for considerably more work.
Please log in or register to report this question.
A locker in bank has 3 digit lock. Mahesh forgot his password and was trying all possible combinations. He took 6 seconds for each try. The problem was that each digit can be from 0 to 9. How much time will be needed to by Mahesh to try all the combinations?
Answer: D
Each of the three digits can be any of 0 to 9, so the digits are independent.
\[ 10 \times 10 \times 10 = 1000 \text{ combinations} \]
\[ 1000 \times 6 = 6000 \text{ seconds} = \frac{6000}{60} = 100 \text{ minutes} \]
Repetition is allowed here - nothing stops a code such as 777 - so it is \( 10^{3} \) rather than \( 10 \times 9 \times 8 \).
Please log in or register to report this question.
On a shelf, 2 books of Geology, 2 books of Sociology and 5 of Economics are to be arranged in such a way that the books of any subject are to be together. Find in how many ways can this be done?
Answer: B
Treat each subject as a block, then arrange within each.
\[ \text{blocks} = 3! = 6 \]
\[ \text{Geology} = 2!, \quad \text{Sociology} = 2!, \quad \text{Economics} = 5! \]
\[ 6 \times 2 \times 2 \times 120 = 2880 \]
The three subject blocks can themselves be ordered in any of six ways along the shelf.
Please log in or register to report this question.
There are 5 tasks and 5 persons, Task 1 cannot be assigned to either person 1 or person 2, Task 2 must be assigned to either person-3 or person-4 every person is to be assigned one task, In how many ways can the assignment be done?
Answer: C
Deal with the most restricted assignment first.
Task 2 must go to person 3 or person 4:
\[ 2 \text{ ways} \]
Task 1 then cannot go to person 1 or 2, and one of persons 3 and 4 is already used, so two people remain eligible for it:
\[ 2 \text{ ways} \]
The other three tasks go to the remaining three people freely:
\[ 3! = 6 \]
\[ 2 \times 2 \times 6 = 24 \]
Assigning the constrained tasks first keeps the later counts independent of which choice was made.
Please log in or register to report this question.
In how many different ways can four books A, B, C and D be arranged one above another in a vertical order such that the books A and B are never in continuous position?
Answer: B
Count all orders, then remove those with A and B together.
\[ \text{total} = 4! = 24 \]
Treating A and B as one block gives three units, and the pair can be either way round:
\[ 3! \times 2! = 6 \times 2 = 12 \]
\[ 24 - 12 = 12 \]
With only four books, exactly half the arrangements keep A and B apart.
Please log in or register to report this question.
In Daya’s bag there are 3 books of History, 4 books of Science and 2 books of Maths. In how many ways can Daya arrange the books so that all the books of same subject are together?
Answer: D
Treat each subject as a single block, then arrange within each block.
\[ \text{blocks among themselves} = 3! = 6 \]
\[ \text{History} = 3! = 6, \quad \text{Science} = 4! = 24, \quad \text{Maths} = 2! = 2 \]
\[ 6 \times 6 \times 24 \times 2 = 1728 \]
Forgetting the \( 3! \) for the blocks themselves would give 288; the subjects can appear in any order on the shelf.
Please log in or register to report this question.
In how many ways can three consonants and two vowels be selected from the letters of the word "TRIANGLE"?
Answer: D
TRIANGLE has consonants T, R, N, G, L and vowels I, A, E.
\[ {}^{5}C_{3} \times {}^{3}C_{2} = 10 \times 3 = 30 \]
Only the selection is asked for, not the arrangement, so combinations apply; arranging the five chosen letters would multiply this by \( 5! \).
Please log in or register to report this question.
