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Combinations and Permutations : Six Problems

1) Give clearly justified answers to the following.
a) How many 7-digit telephone numbers can be formed if the first digit cannot be 0 or 9 and if the last digit is greater than or equal to 2 and less than or equal to 3? Repeated digits are allowed.
b) How many different ways are there to arrange the 6 letters of the word CANDLE?
c)Tim writes the letters of his name on cards (one letter on each card) and puts them in a hat. He mixes up the cards and randomly picks a card out. He returns the card, mixes them up, and picks another card. He repeats this one more time. What is the probability that he picks the letters of his name (in the correct order)?
d)In how many ways can 12 people be placed on 3 distinct teams of 3, 5, and 4 members?
e)A contestant tosses a fair six-sided die. He receives $18 if a 3 appears and pays $3 if a number other than 3 appears. What is the expected value of a trial of this game?
f)suppose that a hand of 8 cards is dealt from a standard deck of cards. What is the probability that the hand has atleast one spade?

Solution Preview

Dear student, though this topic is bit complex but I can give you some tips :
I. for arrangement use permutation = nPr = n!/(n-r)!
II. for selection use combination = nCr = n!/r!(n-r)!
III. r time repeatition (@ by n ways) = n^r
IV. probability = p = favouring cases/total cases
NOTE: some time it is bit difficult to estimate no. of favouring cases. In that case use p = 1 - q = 1 - probability of opposite happening.
V. Wherever word AND comes, means mulitplication theorem to be applied. Whereever OR comes, addition theorem to be applied.

First digit can notbe 0 or 9, hence it can be written by no. of ways = 8
Last digit(>= 2 and <=3), ...

Solution Summary

Combination and permutation problems are solved. The solutions are well explained. General tips are included.