Number of prizes for a degree = (2^bit range)^degree - (2^bit range)^(degree-1) - (2^bit range)^(degree-2) - ...
Number of prizes for a degree = (2^bit range)^degree - (2^bit range)^(degree-1)
Number of prizes for a degree = 2^(bit range * degree) - 2^(bit range * (degree-1))
2 ^ {bitRange \times n} = f(bitRange, n) + f(bitRange, n-1) + f(bitRange, n-2) + ... + f(bitRange, 1) + f(bitRange, 0)
f(bitRange, n) = 2 ^ {bitRange \times n} - ( f(bitRange, n-1) + f(bitRange, n-2) + ... + f(bitRange, 1) + f(bitRange, 0) )
f(bitRange, n) = 2 ^ {bitRange \times n} - f(bitRange, n-1) - ( f(bitRange, n-2) + ... + f(bitRange, 1) + f(bitRange, 0) )

Because:

2 ^ {bitRange \times (n-1)} = f(bitRange, n-1) + f(bitRange, n-2) + ... + f(bitRange, 1) + f(bitRange, 0)
2 ^ {bitRange \times (n-1)} - f(bitRange, n-1) = f(bitRange, n-2) + ... + f(bitRange, 1) + f(bitRange, 0)

Therefore:

f(bitRange, n) = 2 ^ {bitRange \times n} - f(bitRange, n-1) - ( 2 ^ {bitRange \times (n-1)} - f(bitRange, n-1) )
f(bitRange, n) = 2 ^ {bitRange \times n} - f(bitRange, n-1) - 2 ^ {bitRange \times (n-1)} + f(bitRange, n-1)
f(bitRange, n) = 2 ^ {bitRange \times n} - 2 ^ {bitRange \times (n-1)}
f(bitRange, n) = ( 1 << bitRange * n ) - ( 1 << bitRange * (n - 1) )
f(bitRange, n) = (2 ^ {bitRange} - 1) \times 2 ^ {bitRange \times (n - 1)}
f(bitRange, n) = 2 ^ {bitRange} \times 2 ^ {bitRange \times (n - 1)} - 2 ^ {bitRange \times (n - 1)}
f(bitRange, n) = 2 ^ {bitRange + bitRange \times (n - 1)} - 2 ^ {bitRange \times (n - 1)}
f(bitRange, n) = 2 ^ {bitRange + bitRange \times n - bitRange} - 2 ^ {bitRange \times (n - 1)}
f(bitRange, n) = 2 ^ {bitRange \times n} - 2 ^ {bitRange \times (n - 1)}
/**
    * @notice Calculates the number of prizes for a given prizeDistributionIndex
    * @param _bitRangeSize Bit range size for Draw
    * @param _prizeTierIndex Index of the prize tier array to calculate
    * @return returns the fraction of the total prize (base 1e18)
    */
function _numberOfPrizesForIndex(uint8 _bitRangeSize, uint256 _prizeTierIndex)
    internal
    pure
    returns (uint256)
{
    uint256 bitRangeDecimal = 2**uint256(_bitRangeSize);
    uint256 numberOfPrizesForIndex = bitRangeDecimal**_prizeTierIndex;

    while (_prizeTierIndex > 0) {
        numberOfPrizesForIndex -= bitRangeDecimal**(_prizeTierIndex - 1);
        _prizeTierIndex--;
    }

    return numberOfPrizesForIndex;
}
if (_prizeTierIndex > 0) {
    return ( 1 << _bitRangeSize * _prizeTierIndex ) - ( 1 << _bitRangeSize * (_prizeTierIndex - 1) );
} else {
    return 1;
}
