Cool Combinations!

From 33 and 77, it is clear that there are 2 "3" and 2 "7" with 1 "8".

Now the main task is to arrange them.

Sine 78 and 87 both are present, 8 must be in middle of both 7.

$\color{#3D99F6}{\text{\_\_\_\_\_7\_\_\_\_\_\_8\_\_\_\_\_\_\_7\_\_\_\_\_}}$

Now 83 and 38 are present, so 3 will be in opposite side.

$\color{#3D99F6}{\text{\_\_\_\_\_3\_\_\_\_\_\_8\_\_\_\_\_\_\_3\_\_\_\_\_}}$

There is no 73, which implies 7 will be at last and number would start from 3.

$\color{#3D99F6}{\text{3\_\_\_7\_\_\_\_8\_\_\_\_3\_\_\_\_\_7}}$

This ends the puzzle. The number is $\color{#D61F06}{\boxed{37837}}$

Python 2.7:

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pairs_of_N = [33,37,37,37,38,73,77,78,83,87]
def pairs(n):
    digits = str(n)
    n_pairs = []
    for i_1 in xrange(len(digits)):
        for i_2 in xrange(len(digits)):
            if i_1 < i_2:
                n_pairs.append(int(digits[i_1]+digits[i_2]))
    n_pairs.sort()
    return n_pairs
n = 1
while pairs(n) != pairs_of_N:
    n += 1
print "Answer:", n