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1. Among 9 consonants and 5 vowels, what number of words can be formed of 4 consonants and 3 vowels?
6400400
6350400
6550500
6350600
The number of ways to choose 4 consonants from 9 is given by the combination formula:
Number of ways to choose 4 consonants = 9C4 = 9! / (4! * (9-4)!) = 9! / (4! * 5!) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126
Similarly, the number of ways to choose 3 vowels from 5 is given by:
Number of ways to choose 3 vowels = 5C3 = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = (5 * 4) / (2 * 1) = 10
Now, we can combine the chosen consonants and vowels to form words. Since the order matters, we can use the multiplication principle.
Number of words = (Number of ways to choose 4 consonants) * (Number of ways to choose 3 vowels) = 126 * 10 = 1260
Therefore, the number of words that can be formed with 4 consonants and 3 vowels is 1260.
the 7 letters can be arranged in 7! ways :
= 1260 * 7! = 1260 *5040
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2. There are 4 bananas, 7 apples and 6 mangoes present in the fruit basket of kareena kapoor who went to supermarket . In how many ways can a person make a selection of fruits from the basket.
269
256
279
280
Zero or more bananas can be selected in 4 + 1 = 5 ways (0 orange, 1 orange, 2 orange, 3 orange and 4 orange) similarly apples can be selected in 7 +1 = 8 ways and mangoes in 6 +1 = 7 ways so total number of ways = 5*8*7 = 280 but we included a case of 0 orange, 0 apple and 0 mangoes, so we have to subtract this, so 280 – 1 = 279 ways
3. Find out the number of arrangements that can be made for the word RAINBOW if all the the vowels are placed on the even places?
720
144
120
36
The word RAINBOW has 7 letters in which vowels are 3.
As mentioned in the question the vowels have to reside in even places, so they can be set in the 3 even positions in 3! So 3x2x1=6 ways. Also the consonants can be set in the left over 4 positions in 4! So 4x3x2x1=24 ways.
Therefore the total number of ways will be: 24 * 6 = 144.
4. In how many ways the word 'MESMERISE' can be written?
5!/(2!)2 3!
9!/(2!)3 3!
9!/(2!)2 (2!)3
10!/(2!*3!)
Clearly the word 'MESMERISE' contains of 10 letters in which there are M are 2 E are 3, S are 2 and I and R are 1.
So the total number of arrangements are= 10!/(2!*3!.
5. A panel has total of 11 members 5 males and 6 females. Find out the number of ways of picking 2 males and 3 females from the given panel team?
110
300
200
350
The total number of ways to pick two male and three female = ⁵C₂ * ⁶C₃
= 200
6. There are 21 stations between Chennai and Darjeeling. Find out that what is the number of second class tickets have to be produced, so that a traveler can travel from any point of station to the any of the station?
506
380
950
180
The total number of stations = 23
From 23 stations we need to select any 2 stations and the direction of travel (i.e., Chennai to Darjeeling is different from Darjeeling to Chennai) in ²³P₂ ways.
²³P₂ = 23 * 22 = 506.
7. A device password comprises 5 rings each having 7 different messages. Find out the maximum number of different unsuccessful efforts to open the lock?
22316
24321
16806
17283
As each ring have 7 letters. Therefore, the number of tries made with the five rings is = 7 * 7 * 7 * 7 * 7 = 16807. Out of these efforts, one will be successful effort.
Maximum number of unsuccessful efforts = 16807 - 1 = 16806.
8. Find out the number of ways in which 6 male teachers and 6 female teachers can be settled on seat for a shoot so that no two female teachers sit simultaneously?
2 (6!)
6! * ⁷P₆
1(6!) 2
9! * 6
As given first we will arrange all six male teachers in 6! Ways.
Therefore, now 6 female teachers require to be arranged on 7 seats and it can be done in ⁷P₆ ways.
Thus, number of ways = 6! * ⁷P₆
9. By using the digits (1, 3, 4, 5, 6, 7, 8) how many five digit numbers can be made? Rule*(Recurrence of numbers is not allowed)?
3610
6020
2520
2180
We have total of seven digits.
Five digit numbers which can be formed in ways = ⁷P⁵
= 7* 6 * 5 * 4 * 3 = 2520
10. Out of numbers (1, 2,3,5,7 & 9) how many four digit even numbers can be formed?
100
360
60
260
The numbers we have: 1, 2, 3, 5, 7, 9
Out of the given numbers, there is only one even digit number i.e.2. So elements place is occupied with only '2' and the left over 3 positions can be filled as: ⁵P₃ ways.
So the number of even numbers = ⁵P₃ = 60.
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