How to determine maximum positive base-10 value of a float mantissa?

ieee-754 uses significand rather than mantissa.

C does not define mantissa. C uses significand.

Common float normal¹ values have a 24-bit significand that is made up of 1 implied bit with a value of 1 and 23 explicitly encoded binary fractional bits. All 2²⁴ combination are possible.

The maximum significand is 1.11111111 11111111 1111111₂ or 1.99999988079071044921875₁₀ or (2.0-2^-23).

When this is combined with the maximum binary exponent for finite numbers 2^(254-127), the maximum float, FLT_MAX is 340282346638528859811704183484516925440.0 or about 3.402823466e+38.

¹For sub-numerals, there is no implied bit.

That maximum significand is 0.11111111 11111111 1111111₂ or 0.99999988079071044921875₁₀

The number of possible values of a normal significand of a float is (FLT_RADIX-1)/FLT_EPSILON, where FLT_RADIX and FLT_EPSILON are defined by including <float.h>.

This is because FLT_EPSILON is the step size from 1 to the next greater representable number, so it is a change of 1 in the significand bits (when they are interpreted as a binary integer and we are starting from the floating-point number 1.000…000). FLT_RADIX/FLT_EPSILON calculates how many steps the significand could go through, starting from 0, until it wraps or overflows its leading digit. However, we do not start at zero; the question requests excluding the implicit leading 1 bit. The leading bit of a normalized binary-based floating-point number is 1, but, when we generalize to other bases, the leading digit of a floating-point number may be something other than 1 for a normalized number; it can be a non-zero integer less than FLT_RADIX. So, starting from 1 instead of 0, there are (FLT_RADIX-1)/FLT_EPSILON possible values of normal significands.

Note that (FLT_RADIX-1)/FLT_EPSILON has an integer value but floating-point type. To use it as an integer type, you may need a cast, such as when printing it with %d.

The floating-point number with the same scale (exponent) as 1 but the maximum significand is FLT_RADIX - FLT_EPSILON. The maximum value of the significand as an integer is FLT_RADIX/FLT_EPSILON - 1. Note that the latter includes the leading digit.

Notes

“Significand” is the preferred term for the fraction portion of a floating-point number. “Mantissa” is an old term for the fraction portion of a logarithm. Significands are linear; multiplying a significand multiplies the number represented. Mantissas are logarithm; adding to a mantissa multiplies the number represented.

“Permutation” refers to moving things around; (1 2 3 4) and (3 4 2 1) are permutations of each other. You appear to want the number of different values the significand bits can have.

The magnitude of the mantissa has no meaning without considering the exponent. What the 23 tells you is that the number of significant decimal digits is 23 * log(2) ≈ 7.

However a 24th bit is implied, giving 24 * log(2) which is > 7. So all 7-digit integer values can be stored without loss of precision.

Also, any integer that has a power of 2 as a factor, and when divided by that factor has 7 digits or less, can also be exactly represented, as the power of 2 is taken up by the exponent (subject to the limit of the exponent value).

So it is the exponent size that gives the range of values that can be stored, while the mantissa (significand) size gives the precision.