Advertisements
Advertisements
Question
There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:
a particular student is included.
Advertisements
Solution
Clearly, 2 professors and 3 students are selected out of 10 professors and 20 students, respectively.
Required number of ways =\[{}^{10} C_2 \times^{20} C_3 = \frac{10}{2} \times \frac{9}{1} \times \frac{20}{3} \times \frac{19}{2} \times \frac{18}{1} = 51300\]
If a particular student is included, it means that 2 students are selected out of the remaining 19 students.
Required number of ways =\[{}^{19} C_2 \times^{10} C_2 = \frac{19}{2} \times \frac{18}{1} \times \frac{10}{2} \times \frac{9}{1} = 7695\]
APPEARS IN
RELATED QUESTIONS
Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.
In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?
It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible?
Prove that
A coin is tossed five times and outcomes are recorded. How many possible outcomes are there?
Given 7 flags of different colours, how many different signals can be generated if a signal requires the use of two flags, one below the other?
How many A.P.'s with 10 terms are there whose first term is in the set {1, 2, 3} and whose common difference is in the set {1, 2, 3, 4, 5}?
From among the 36 teachers in a college, one principal, one vice-principal and the teacher-incharge are to be appointed. In how many ways can this be done?
How many four-digit numbers can be formed with the digits 3, 5, 7, 8, 9 which are greater than 7000, if repetition of digits is not allowed?
Serial numbers for an item produced in a factory are to be made using two letters followed by four digits (0 to 9). If the letters are to be taken from six letters of English alphabet without repetition and the digits are also not repeated in a serial number, how many serial numbers are possible?
A number lock on a suitcase has 3 wheels each labelled with ten digits 0 to 9. If opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? Also, find the number of unsuccessful attempts to open the lock.
Evaluate the following:
n + 1Cn
Evaluate the following:
If nC12 = nC5, find the value of n.
f 24Cx = 24C2x + 3, find x.
If 15Cr : 15Cr − 1 = 11 : 5, find r.
If nC4 , nC5 and nC6 are in A.P., then find n.
From 4 officers and 8 jawans in how many ways can 6 be chosen. to include at least one officer?
Find the number of diagonals of , 1.a hexagon
Find the number of (ii) triangles
Determine the number of 5 cards combinations out of a deck of 52 cards if at least one of the 5 cards has to be a king?
A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: exactly 3 girls?
In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?
A business man hosts a dinner to 21 guests. He is having 2 round tables which can accommodate 15 and 6 persons each. In how many ways can he arrange the guests?
There are 3 letters and 3 directed envelopes. Write the number of ways in which no letter is put in the correct envelope.
If mC1 = nC2 , then
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is
There are 20 straight lines in a plane so that no two lines are parallel and no three lines are concurrent. Determine the number of points of intersection.
Ten students are to be selected for a project from a class of 30 students. There are 4 students who want to be together either in the project or not in the project. Find the number of possible selections.
Answer the following:
A question paper has 6 questions. How many ways does a student have to answer if he wants to solve at least one question?
In how many ways can the letters of the word 'IMAGE' be arranged so that the vowels should always occupy odd places?
A student has to answer 10 questions, choosing atleast 4 from each of Parts A and B. If there are 6 questions in Part A and 7 in Part B, in how many ways can the student choose 10 questions?
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated.
If 20 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, in how many points will they intersect each other?
A box contains 2 white balls, 3 black balls and 4 red balls. The number of ways three balls be drawn from the box if at least one black ball is to be included in the draw is ______.
Eighteen guests are to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on other side of the table. The number of ways in which the seating arrangements can be made is `(11!)/(5!6!) (9!)(9!)`.
There are 12 balls numbered from 1 to 12. The number of ways in which they can be used to fill 8 places in a row so that the balls are with numbers in ascending or descending order is equal to ______.
