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\[\text{ A sequence } x_0 , x_1 , x_2 , x_3 , . . . \text{ is defined by letting } x_0 = 5 and x_k = 4 + x_{k - 1}\text{ for all natural number k . } \]
\[\text{ Show that } x_n = 5 + 4n \text{ for all n } \in N \text{ using mathematical induction .} \]
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\[\text{ The distributive law from algebra states that for all real numbers} c, a_1 \text{ and } a_2 , \text{ we have } c\left( a_1 + a_2 \right) = c a_1 + c a_2 . \]
\[\text{ Use this law and mathematical induction to prove that, for all natural numbers, } n \geq 2, if c, a_1 , a_2 , . . . , a_n \text{ are any real numbers, then } \]
\[c\left( a_1 + a_2 + . . . + a_n \right) = c a_1 + c a_2 + . . . + c a_n\]
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In how many ways can 4 letters be posted in 5 letter boxes?
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Write the number of 5 digit numbers that can be formed using digits 0, 1 and 2 ?
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In how many ways 4 women draw water from 4 taps, if no tap remains unused?
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Write the total number of possible outcomes in a throw of 3 dice in which at least one of the dice shows an even number.
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Write the number of arrangements of the letters of the word BANANA in which two N's come together.
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Write the number of ways in which 7 men and 7 women can sit on a round table such that no two women sit together ?
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Write the number of words that can be formed out of the letters of the word 'COMMITTEE' ?
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Write the number of all possible words that can be formed using the letters of the word 'MATHEMATICS'.
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Write the number of ways in which 6 men and 5 women can dine at a round table if no two women sit together ?
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Write the number of ways in which 5 boys and 3 girls can be seated in a row so that each girl is between 2 boys ?
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Write the remainder obtained when 1! + 2! + 3! + ... + 200! is divided by 14 ?
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Write the number of numbers that can be formed using all for digits 1, 2, 3, 4 ?
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The number of permutations of n different things taking r at a time when 3 particular things are to be included is
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The number of five-digit telephone numbers having at least one of their digits repeated is
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The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is
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How many numbers greater than 10 lacs be formed from 2, 3, 0, 3, 4, 2, 3 ?
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The number of different signals which can be given from 6 flags of different colours taking one or more at a time, is
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