Find how many solutions there are to the given equations a)...

Find how many solutions there are to the given equations a) x + y + z = 20, where x,y,z are nonnegative integers b) x + y + z = 20, where x,y,z is a positive integer c) a + b + c + d = 30, where a,b,c,d are non negative integers d) a + b + c + d = 30, where a,b,c,d are integers that are at least two e) a + b + c + d + e = 500, where a,b,c,d,e are integers that are at least ten

Transcript

00:01 Hi, in this question we will use the formula to find the number of solutions as n plus r minus 1c r to find the number of solutions for the given equation.

00:20 So starting with the first one, we are given with x plus y plus z is equals to 20 and x y z are non -negative integers.

00:30 That means we are having n is equal to 3, r is equal to 20.

00:35 So substituting this we get 3 plus 20 minus 1 c20 which is equals to 22 c20.

00:45 Or solving this further, we get this to be equal to 22 into 21 into 20 factorial divided by 20 factorial into 22 minus 20 that is 2 factorial.

00:59 This will get cancelled and solving this further we get 22 into 21 by 2 into 1.

01:06 Here this will again get cancelled and we will left with the value that is 231.

01:13 Now moving on to the second part we are given with x plus y plus z is equal to 20.

01:20 Here in this we are given that here x y and z is a positive integer.

01:26 That means x y and z they are greater than one so we will let the value for a is equals to x minus 1 b is equals to y minus 1 and c is equal to z minus 1 or we can get the value for x which is a plus 1 y which is b plus 1 and z which is c plus 1 substituting this into the given equation we get a plus 1 plus b plus 1 plus c plus 1 plus 1 is equal to 20 or we can write a plus b plus c is equal to 17.

02:03 Now finding the number of solution here we can see we have n is equal to 3 and r is equal to 17.

02:10 So substituting again we get the result as n plus r that is 3 plus 17 minus 1 c 17 or we get this to be equal to 19 c 17 solving this further we get 19 into 18.

02:26 18 into 17 factorial divided by 17 factorial into 2 factorial.

02:32 Here this and this would get cancelled.

02:34 We are left with 19 into 18 by 2 into 1.

02:38 This will again get cancelled and we'll left with the result 171.

02:44 Now moving on to the third part we are given with the equation a plus b plus c plus d is equal to 30 and we are given a, b, c are non -negative integers.

02:56 So we can see here, n is equal to 4, r is equal to 30.

03:03 So substituting the value into the given relation, max c 30, or we get this to be equal to 33 c 30.

03:14 Solving this further, we get 33 into 32, into 31 into 30 factorial, divided by 30 factorial into 3 factorial.

03:25 This and this would get cancelled.

03:26 And we will left with 33 into 32 into 31 divided by 3 into 2 into 1...

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