Each major has its own admission pathways and requirements. You may wish to confirm using the link below, but last time I checked, the majors that have a “direct admission” pathway are business administration, music and other audition based majors, and some of the engineering majors.
“This is a list of UW-Seattle undergraduate majors. Options and concentrations are listed when available. Each major links to the General Catalog, where you can learn more about its admission and graduation requirements.”
http://www.washington.edu/uaa/advising/academic-planning/majors-and-minors/list-of-undergraduate-majors/
Here is some general guidance regarding majors:
http://www.washington.edu/uaa/advising/academic-planning/majors-and-minors/overview/
Regarding the engineering majors, I do not know how many applicants were offered a direct admission to an engineering major last year, but according to the ASEE Online Profile, around 15.1% of freshmen in the College of Engineering were enrolled in a major (the others were enrolled with pre-engineering status).
Regarding computer science and engineering, a recent article in the Seattle Times about Paul Allen’s $40 million gift to the establish the new Allen School for Computer Science & Engineering reported:
"Students are usually admitted at the end of their freshman year or the start of their sophomore year. But the school also admits some students directly, at the same time they are admitted to the university as freshmen. This year, direct admission will be offered to 150 students. All but five of them are Washington residents.
"Although computer science and engineering is a tough program to get into, the picture is getting better. In 2016, 43 percent of bachelor’s applicants were admitted to the program. In previous years, the admission rate was lower.
“[The CSE Department] is now enrolling about 370 new undergraduate students each year, double the number it was able to accommodate in 2012… This year, it is asking for $6 million from the Legislature to increase the number of students it graduates by 120 degrees per year. Most of that increase would be in bachelors degrees, Lazowska said.”