If x, y, z, t are odd natural numbers such that `x + y + z +t=20` then find the number of values of ordered quadruplet (x, y, z, t).

Explanation

To solve the problem of finding the number of ordered quadruplets (x, y, z, t) of odd natural numbers such that \( x + y + z + t = 20 \), we can follow these steps:

### Step 1: Express odd natural numbers in a different form
Since \( x, y, z, t \) are odd natural numbers, we can express them in the form:
- \( x = 2p + 1 \)
- \( y = 2q + 1 \)
- \( z = 2r + 1 \)
- \( t = 2s + 1 \)

where \( p, q, r, s \) are non-negative integers.

### Step 2: Substitute into the equation
Substituting these expressions into the equation \( x + y + z + t = 20 \):
\[ (2p + 1) + (2q + 1) + (2r + 1) + (2s + 1) = 20 \]

### Step 3: Simplify the equation
This simplifies to:
\[ 2p + 2q + 2r + 2s + 4 = 20 \]
Subtracting 4 from both sides gives:
\[ 2p + 2q + 2r + 2s = 16 \]

### Step 4: Divide by 2
Dividing the entire equation by 2 results in:
\[ p + q + r + s = 8 \]

### Step 5: Find the number of non-negative integer solutions
Now, we need to find the number of non-negative integer solutions to the equation \( p + q + r + s = 8 \). This can be solved using the "stars and bars" theorem, which states that the number of solutions to the equation \( x_1 + x_2 + ... + x_k = n \) in non-negative integers is given by:
\[ \binom{n + k - 1}{k - 1} \]
where \( n \) is the total sum and \( k \) is the number of variables.

In our case:
- \( n = 8 \)
- \( k = 4 \) (for \( p, q, r, s \))

### Step 6: Apply the formula
Using the formula:
\[ \text{Number of solutions} = \binom{8 + 4 - 1}{4 - 1} = \binom{11}{3} \]

### Step 7: Calculate the binomial coefficient
Calculating \( \binom{11}{3} \):
\[ \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = \frac{990}{6} = 165 \]

### Conclusion
Thus, the number of values of the ordered quadruplet \( (x, y, z, t) \) is \( 165 \).

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