1. How many words of length 5 are there on the set {A, B, C, D} which do NOT containthe string 'ADC'? For example, ADÇDD, BADÇB are not counted but DAÇAD iscounted.2. Let n, r e N. In how many ways can r distinct balls be placed into n distinct boxes?3. In how many ways can words of length 5 be made from the alphabets in the Englishlanguage, so that no vowel appears in the even positions of the word?4. Determine the number of possible outcomes, if three distinct coins are tossed and fivedistinct dice are rolled?5. How many functions f : [6] → [5] satisfy the condition f(i) + i for at least two values ofi in {1,2, ..., 6}?6. Determine the number of 5 digit positive numbers, such that the digit 9 does not appearexactly 4 times in the number?Section 2:1. Suppose we wish to arrange 5 people {A, B, C, D, E}, standing side by side, for a pho-tograph. How many such (distinct) photographs are possible? Determine the number ofdistinct photographs with exactly 3 people from the above set.2. Determine the mumber of one-one maps from [4] = {1,2,3, 4} to the set of all the 26alphabets {A, B, .., Z}.3. How many words can be obtained by rearranging all the letters of the word ROY AL?4. How many one-one and onto maps f : [12] → [12] exist if the image of a multiple of 3 isalways a multiple of 3?Section 3:1. Find the largest positive integer n such that 2" divides 35! but 2"+1 does not divide 35!.2 II then the value of r equals ....3. There are 5 doors in a Lecture Hall. In how many ways a student can enter the roomthrough one door and leave through another? What if he is also allowed to leave throughthe door that he has entered?4. In an exam there are 6 questions, each having 4 choices and only one of them beingCORRECT. Each student has answered the question paper. If no student has got allcorrect choices and no two student have written the same sequence of answers then whatis the maximum number of students in the class, for the above to happen?5. How many numbers between 100 and 1000(a) have 2 at the unit place?(b) have 2 as one of the dligits?(c) have exactly two numbers the same?(d) are divisible by 16?6. In how many ways the letters of the word "HAPPYWENTTOANUNHAPPYPARTY" bearranged so that the first appearance of the letter "H" occurs before the first appearanceof "T"?

Question

1. How many words of length 5 are there on the set {A, B, C, D} which do NOT containthe string 'ADC'? For example, ADÇDD, BADÇB are not counted but DAÇAD iscounted.2. Let n, r e N. In how many ways can r distinct balls be placed into n distinct boxes?3. In how many ways can words of length 5 be made from the alphabets in the Englishlanguage, so that no vowel appears in the even positions of the word?4. Determine the number of possible outcomes, if three distinct coins are tossed and fivedistinct dice are rolled?5. How many functions f : [6] → [5] satisfy the condition f(i) + i for at least two values ofi in {1,2, ..., 6}?6. Determine the number of 5 digit positive numbers, such that the digit 9 does not appearexactly 4 times in the number?Section 2:1. Suppose we wish to arrange 5 people {A, B, C, D, E}, standing side by side, for a pho-tograph. How many such (distinct) photographs are possible? Determine the number ofdistinct photographs with exactly 3 people from the above set.2. Determine the mumber of one-one maps from [4] = {1,2,3, 4} to the set of all the 26alphabets {A, B, .., Z}.3. How many words can be obtained by rearranging all the letters of the word ROY AL?4. How many one-one and onto maps f : [12] → [12] exist if the image of a multiple of 3 isalways a multiple of 3?Section 3:1. Find the largest positive integer n such that 2&#34; divides 35! but 2&#34;+1 does not divide 35!.2 II then the value of r equals ....3. There are 5 doors in a Lecture Hall. In how many ways a student can enter the roomthrough one door and leave through another? What if he is also allowed to leave throughthe door that he has entered?4. In an exam there are 6 questions, each having 4 choices and only one of them beingCORRECT. Each student has answered the question paper. If no student has got allcorrect choices and no two student have written the same sequence of answers then whatis the maximum number of students in the class, for the above to happen?5. How many numbers between 100 and 1000(a) have 2 at the unit place?(b) have 2 as one of the dligits?(c) have exactly two numbers the same?(d) are divisible by 16?6. In how many ways the letters of the word &#34;HAPPYWENTTOANUNHAPPYPARTY&#34; bearranged so that the first appearance of the letter &#34;H&#34; occurs before the first appearanceof &#34;T&#34;?

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1. How many words of length 5 are there on the set {A, B,C, D} which do NOT contain the string 'ADC'? For example, ADCDD, BADCB are not counted but DACAD is counted.