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Greatest terms in Combinations

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Question 11

Question: There are 8 objects in column A and 8 objects in column B. A boy is asked to match each item in column A with each item in column B. How many possible (correct or wrong) answers are there for these questions?

Answer: Each answer to this question is an arrangement of 8 items of column
B so as to correspond to the given arrangement of items in column A.
Therefore, there are as many arrangements as there are answers.
Therefore, number of answers required = P(8,8) = 8!

Question 12

Question: Find n; r if

Answer:

n = 10 is only valid.

Question 13

Question: If there are 8 periods in each working day in a school, in how many ways can 6 subjects be arranged such that each subject is allowed atleast one period?

Answer: Out of 8 periods, 6 periods be arranged in ⁸P₆ ways and the remaining 2 periods be arranged in ⁶P₂ ways.

Question 14

Question: Four alphabets T, A, L and K, one each were purchased from a plastic warehouse. How many ordered pairs of alphabets to be used as initials, can be formed from them?

Answer: Four letters A, K, L, T are purchased. For an initial, two alphabets are required.
Thus, the number of initials that can be formed is P (4,2) = 4x3 = 12.
The number of initials that can be formed are 12.

Question 15

Question: How many three digit numbers are there with distinct digits with each digit even, i) with 0 and ii) without 0?

Answer: Here we are required to form three-digit numbers with distinct each digit even.
The even digits are 0, 2, 4, 6, and 8.

The digits used are 0, 2, 4, 6, and 8.
The hundred's place can be filled in 4 ways (excluding zero).
The ten's place can be filled in again in 4 ways (including zero).
The unit's place can be filled in 3 ways.
Hence, by the fundamental principle of counting, the number of three
digit numbers = 4 x 4 x 3 = 48.

ii) Without 0
The digits used are 2, 4, 6, and 8.
The hundred's place can be filled in 4 ways.
The ten's place can be filled in 3 ways and the unit's place can be filled
in 2 ways.
Hence, by the fundamental principle of counting, the number of three digit numbers = 4 x 3 x 2 = 24.

Question 16

Question: m men and n women are to be seated in a row so that no two women sit together. If m>n, then show that the number

Answer: m men can be arranged in m seats in m! ways. After the m men have been seated, we may get one seat in the beginning and one seat in the end and (m-1) seats in between each pair of men for the women.
Thus, the number of ways in which n women can be seated in

Question 17

Question: Ten articles are to be placed in ten boxes, one in each box. Five of them are too big for four boxes. Find the number of possible arrangements.

Answer: 5 of the ten articles which are too big for 4 boxes are to be kept in
the remaining 6 boxes. This can be done in ⁶P₅ ways. Now there are 4
boxes and 5 articles left. This can be arranged in ⁵P₅ ways.
Since, these two events are independent the total number of

Question 18

Question: How many permutations can be made out of the letters of the word TRIANGLE? How many of these begin with i and end with e?

Answer: i) There are 8 distinct letters in the word TRIANGLE. The number of permutations is P(8,8) = 8! = 40320

'i' is the first letter and 'e' is the last letter of each word. The remaining places can be filled in 6! ways = 720 ways.

Question 19

Question: Find the number of arrangements that can be made out of the letters of the words:
i) INDEPENDENCE ii) SUPERSTITIOUS iii) INSTITUTIONS.

Answer: i) In the word INDEPENDENCE, I=1, D's=2, E's=4, N's=3, C=1, P=1.
\Total number of letters = 12

ii) The word SUPERSTITIOUS consists of S's=3, U's=2, P=1, E=1, R=1, T's=2, I's=2, O=1.
\Total number of letters = 13

iii) The word INSTITUTIONS consists of I's=3, N's=2, S's=2, T's=3, O=1, U=1.
\Total number of letters = 12