QBM078
Q1. How many number of times will the digit ‘7' be written when listing the integers from 1 to 1000?
Q2. How many three–digit numbers can be
formed from the digits 2, 3, 4, 5, 6, 7 if
a. There are no restrictions?
b. The numbers formed must all be even?
c. The numbers formed must all be even and no digit may be repeated?
Q3. How many different signals can be made by using at least three distinct flags if there are five different flags from which to select?
Q4. Find the number of ways to draw a straight, (suit does not matter) beginning with a 4 and ending with a 8?
Q5. There are ten subjects in BSUP High School but the sixth standard students have only 5 periods in a day. In how many ways can we form a time-table for the day for the sixth standard students?
Q6. What is the value of 1*1! + 2*2! + 3!*3! + ............ n*n!, where n! means n factorial or n(n-1)(n-2)...1
Q7. In how many ways can 10 identical marbles be distributed among 6 children so that each child gets at least 1 marble?
Q8. A tourist wanted to visit three or more five major cities in India . In how many ways can he plan his tour such that one particular city is always included?
Q9. A university student must take a modern language, a natural science, a social science, and English. If there are four different modern languages, five natural sciences, three social sciences , but each student must take the same English course, how many different ways can the student select his course of study?
Q10. In how many ways can the letters of the word MANAGEMENT be rearranged so that the two As do not appear together?
Q11. Five men check their coats at a wedding reception but lose their tickets. If the coats are handed out in a random way, what is the probability that each gets his own coat?
Q12. How many numbers less than 700 have no repetitions of digits?
Q13. Five students walk into a French classroom with 10 desks. How many different seating arrangements are possible?
Q14. There are 5 Rock songs, 6 Carnatic songs and 3 Indi pop songs. How many different albums can be formed using the above repertoire if the albums should contain at least 1 Rock song and 1 Carnatic song?
Q15. If Sita and Geeta cannot sit next to each other in a row of 6 chairs, in how many ways can Sita and Geeta along with 8 other friends be seated?
Please tell me how to solve question number 6 and 7
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Q6. What is the value of 1*1! + 2*2! + 3!*3! + ............ n*n!, where n! means n factorial or n(n-1)(n-2)...1
ans:you can express the above expression as:
(2-1)*1! + (3-1)*2! + (4-1)*3! +................+(n+1-1)*n!
it can be broken up as:
2! - 1! + 3! - 2! +4! - 3! ....................... n! - (n-1)! + (n+1)! - n!
-1! + (n+1)! = (n+1)! - 1
this will b the ans,if any other clarification neeeded then reply
7 does not occur in 1000. So we have to count the number of times it appears between 1 and 999. Any number between 1 and 999 can be expressed in the form of xyz where 0 < x, y, z < 9.
1. The numbers in which 7 occurs only once. e.g 7, 17, 78, 217, 743 etc
This means that 7 is one of the digits and the remaining two digits will be any of the other 9 digits (i.e 0 to 9 with the exception of 7)
You have 1*9*9 = 81 such numbers. However, 7 could appear as the first or the second or the third digit. Therefore, there will be 3*81 = 243 numbers (1-digit, 2-digits and 3- digits) in which 7 will appear only once.
In each of these numbers, 7 is written once. Therefore, 243 times.
2. The numbers in which 7 will appear twice. e.g 772 or 377 or 747 or 77
In these numbers, one of the digits is not 7 and it can be any of the 9 digits ( 0 to 9 with the exception of 7).
There will be 9 such numbers. However, this digit which is not 7 can appear in the first or second or the third place. So there are 3 * 9 = 27 such numbers.
In each of these 27 numbers, the digit 7 is written twice. Therefore, 7 is written 54 times.
3. The number in which 7 appears thrice - 777 - 1 number. 7 is written thrice in it.
Therefore, the total number of times the digit 7 is written between 1 and 999 is 243 + 54 + 3 = 300
n/a
Q1. How many number of times will the digit ‘7' be written when listing the integers from 1 to 1000?
ANS:
1 digit number : 1
2 digit number : 10 + 9 = 19
3 digit numbers : 10x10(7at units place)+ 9 x 10 (7 at tenth place) + 9 x 10 (7 at hundredth place) = 180
Total = 200