Permutations

How many permutations can be formed by the letters of the word ‘VOWELS’, when (i) there is no restriction on letters; (ii) each word begins with E; (iii) each word begins with O and ends with L; (iv) all vowels come together; (v) all consonants come together?

Answer : (i) There is no restriction on letters The word VOWELS contain 6 letters. The permutation of letters of the word will be 6! = 720 words. Each word begins with Here the position of letter E...

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A customer forgets a four-digit code for an automated teller machine (ATM) in a bank. However, he remembers that this code consists of digits 3, 5, 6, 9. Find the largest possible number of trials necessary to obtain the correct code.

Answer : Given: code consists of digits 3, 5, 6, 9. To find: the largest possible number of trials necessary to obtain the correct code. The customer remembers that this 4 digit code consists of...

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For a set of five true or false questions, no student has written the all correct answer and no two students have given the same sequence of answers. What is the maximum number of students in the class for this to be possible?

Answer : Given: a set of five true – false questions. To find: the maximum number of students in the class. Condition: no student has written the all correct answer and no two students have given...

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A mint prepares metallic calendars specifying months, dates and days in the form of monthly sheets (one plate for each month). How many types of February calendars should it prepare to serve for all the possibilities in the future years?

Answer : To find: types of February calendars that can be prepared. There are two factors to develop FEBRUARY metallic calendars The day on the start of the year of which possibility=7 Whether the...

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Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with total award money of ₹ 1,600. School B wants to spend ₹ 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is ₹ 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award. HINT: By the given data, we have
3 x+2 y+z=1600
4x+y+3 z=2300
x+y+z=900

Solution: Assume the amount considered for sincerity, truthfulness and helpfulness are $x, y$ and $z$ respectively. As per the questions, $3 x+2 y+z=1600$ $\begin{array}{l} 4 x+y+3 z=2300 \\...

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An amount of ₹ 5000 is put into three investments at 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹358. If the total annual income from first two investments is ₹70 more than the income from the third, find the amount of each investment by the matrix method. HINT: Let these investments be ₹x, ₹y and ₹z, respectively. Then, x+y+z=5000, \ldots (i) \begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8 z}{100}=358 \Rightarrow \\ 6 x+7 y+8 z=35800 \ldots (ii) \end{array} And, \frac{6 x}{100}+\frac{7 y}{100}=\frac{8 z}{100}+70 \Rightarrow 6 x+7 y-8 z=7000 . \ldots \text { (iii) }

Solution: Suppose the investments are $\mathrm{x} \mathrm{x}$, Fy and $\mathrm{F} \mathrm{z}$, respectively. Therefore, $x+y+z=5000$ $\begin{array}{l} \frac{6 x}{100}+\frac{7 y}{100}+\frac{8...

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The cost of 4 kg potato, 3 kg wheat and 2 kg of rice is ₹ 60. The cost of 1 kg potato, 2 kg wheat and 3 kg of rice is ₹45. The cost of 6 kg potato, 2 kg wheat and 3 kg of rice is ₹70. Find the cost of each item per kg by matrix method.

Solution: Suppose the price of 1kg potato, wheat and rice is $x$, $y$ and $z$ respectively. As per the question, $4x + 3y + 2z = 60$ $x+ 2y + 3z = 45$ $6x + 2y + 3z = 70$ Now converting the...

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The sum of three numbers is 2. If twice the second number is added to the sum of first and third, we get 1. On adding the sum of second and third numbers to five times the first, we get 6. Find the three numbers by using matrices.

Solution: Assume the numbers are $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$. As per the question, $\begin{array}{l} x+y+z=2 \\ x+2 y+z=1 \\ 5 x+y+z=6 \end{array}$ Now converting the following...

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