Why only 16 Jungian types? Combinatorially it should be 2 * 4! = 48
There are not 16 Jungian types. There are 16 MBTI combinations and 16 Kersian (sub-)temperament types. All are related and easily confused.

Jung, in Psychological Types (Collected Works, Book 6), defines two Attitude Types and four functions, splitting the latter into two groups of two.

Attitude Type is Extraversion and Introversion and takes up 9 of the 10 chapters in the work. In the 10th chapter, he explains the four functions: Sensation, Intuition, Thinking, and Feeling. The first two are irrational functions, the second two, rational functions. This makes a total of three groups of two. Though, Jung treats all four functions as one group, it is helpful to remember they can be bifurcated thusly.

Of both the two Attitude Types and the four functions, one is preferred from each. That is, everyone has all the types and functions, but everyone prefers one from each group.

Chapter 10 then explains how the functions play out within the two Attitude Types, leading to 8 explanations, which are Extraversion and Introversion with each of the four functions. He also explains the role of the secondary function. The secondary function will be the opposite of the primary function. That is, if the primary is irrational, the secondary will be rational, and vice-versa.

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Before Jung published Psychological Types, but based on his works, Meyers-Briggs created the MBTI (Meyers-Briggs Type inventory), a personality inventory based on Jung's functions. The first letter is the preferred Attitude Type, the second letter is the preferred irrational function, the third letter is the preferred rational function, and the fourth letter indicates which function is preferred to be extraverted. Note, the fourth letter does not indicate the primary function. It indicates whether the preferred extraverted function is irrational or rational, which is the inverse of the introvert's primary function.

The combination of one of two Attitude Types, one of two irrational functions, one of two rational functions, and one of two preferred extraverted functions, makes 16 combinations.

In Gifts Differing, Meyers-Briggs defined four function groups (ST, SF, NT, NF) which are like groups to the 16 types.

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Keirsey, working on Temperament theory (reactions to the external environment), found a connection between his work and the MBTI. Note, he did not find a connection between Temperament and Jung's theory. Indeed, he could not, because Jung did not have the fourth letter defined by the MBTI, which is integral to the connection Keirsey found. He also grouped the letters differently than the MBTI.

Keirsey's groups are SP, SJ, NF, NT, which can be matched to the four historical temperaments: Artisan, Guardian, Idealist, Rational. The history of Temperament theory gives many different names for these groups, so Keirsey, in his main work, Please Understand Me II, says he based it on The Republic. In Keirsey's last work, Personology, however, he focuses much less on those names and uses his own.

With a connection between the MBTI and Temperament theory established, Keirsey extended Temperament theory to include four sub-types for each of the four temperaments. This makes a total of 16 sub-types.