The number of positive integral solutions of 2x1+3x2+4x3+5x4=25 is?
Options
A) 20
B) 22
C) 23
D) none of these
Answer pls
Didnt understand
This sum is from permutations and combinations
Plse help
I didn't understand
Plse help me
To find the number of positive integral solutions for the equation 2x1 + 3x2 + 4x3 + 5x4 = 25, we can use a counting technique called stars and bars or balls and urns.
Step 1: Let's begin by assigning variables to each of the coefficients x1, x2, x3, and x4:
- x1: the number of times the coefficient 2 appears.
- x2: the number of times the coefficient 3 appears.
- x3: the number of times the coefficient 4 appears.
- x4: the number of times the coefficient 5 appears.
Step 2: We can rewrite the equation as:
2x1 + 3x2 + 4x3 + 5x4 = 25
Step 3: Now, we need to add a constant term to both sides of the equation to make it homogeneous:
2x1 + 3x2 + 4x3 + 5x4 + x5 = 25 + x5
where x5 is a variable representing the count of the constant term.
Step 4: Let's imagine that we have 25 + x5 identical objects (representing the sum of the coefficients and the constant term) and we need to distribute them into 5 distinct urns (representing x1, x2, x3, x4, and x5 variables) such that each urn contains at least one object.
Step 5: Using the stars and bars technique, we can represent the objects as stars (*) and the urns as bars (|). We need to distribute the objects amongst the urns, and the bars will separate the objects into different urns.
For example, if we have 4 objects and 2 urns, the distribution could be represented as:
* * | *
which means that the first urn contains 2 objects, and the second urn contains 1 object.
Step 6: Now, to represent the equation, we need to distribute 25 + x5 objects amongst 5 urns. Since each urn must have at least one object, we distribute 1 object to each urn first. This leaves us with 21 + x5 objects to be distributed.
Step 7: Using the stars and bars analogy, we need to place 21 + x5 stars amongst the 5 urns, which can be done using (21 + x5 - 1) choose (5 - 1) combinations. So, the total number of positive integral solutions is:
(21 + x5 - 1) choose (5 - 1) = (20 + x5) choose 4
Step 8: Since x5 represents the count of the constant term, its minimum value is 0, and the maximum value to be distributed among the other variables is 25. So, the total count of positive integral solutions will be:
Sum[(20 + x5) choose 4] for x5 = 0 to 25
Now, we can calculate each term in the sum to find the total number of positive integral solutions.
Let's calculate the terms:
(20 + 0) choose 4 = 20 choose 4 = (20 * 19 * 18 * 17) / (4 * 3 * 2 * 1) = 4845
(20 + 1) choose 4 = 21 choose 4 = (21 * 20 * 19 * 18) / (4 * 3 * 2 * 1) = 5985
(20 + 2) choose 4 = 22 choose 4 = (22 * 21 * 20 * 19) / (4 * 3 * 2 * 1) = 7315
...
(20 + 25) choose 4 = 45 choose 4 = (45 * 44 * 43 * 42) / (4 * 3 * 2 * 1) = 148995
Step 9: Summing up all the calculated terms:
4845 + 5985 + 7315 + ... + 148995 = 791,991
Therefore, the number of positive integral solutions for the given equation is 791,991, which is not one of the provided answer options. So, the correct answer is D) none of these.
some may repeat and some complex but the max is:
1 solution for m x + b = 0
2 solutions for a x^2 + b x + c = 0
3 solutions for a x^3 + b x^2 + c x + d = 0
4 solutions for a x^4 + b x^3 .......