The intention is to form a committee consisting of 5 persons from BIM, a group of engineers and technicians. The number of engineers is 10 and the number of technicians is 4. Find the number of possible ways to form a committee to contain at least one technician if the selection is rand
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
The intention is to form a committee consisting of 5 persons from BIM, a group of engineers and technicians. The number of engineers is 10 and the number of technicians is 4. Find the number of possible ways to form a committee to contain at least one technician if the selection is rand
There are 10 engineers and 4 technicians.
Thus, total number of individuals in the population is, n = 14 (= 10 + 4).
Out of these individuals, r = 5 distinct members are to be chosen for the committee.
The number of ways in which the 5 members can be chosen from the 14 engineers and technicians is 14C5 = 2002.
Thus, the total number of ways in which the 5 member-committee can be formed, without any constraints, is 2,002.
Now, the committee consists of at least one technician. The possibilities are:
- 4 engineers and 1 technician
- 3 engineers and 2 technician
- 2 engineers and 3 technician
- 0 engineers and 4 technician
Now, the number of ways in which the 5 member-committee can have at least 1 technician is:
(10C4) (4C1) + (10C3) (4C2) + (10C2) (4C3) + (10C1) (4C4) = 840 + 720 + 180+10=1750
Thus, the number of ways in which the 5 member-committee can have at least 1 technician is 1750.
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