Я встретился с четверыми оценщиками.

Questions & Answers about Я встретился с четверыми оценщиками.

Why use четверыми instead of the standard numeral четырьмя?
четверыми is the Instrumental case of the collective numeral четверо (a group of four). While you could grammatically use the standard cardinal numeral четырьмя, collective numerals are frequently preferred when talking about groups of men or mixed-gender groups of people. Since оценщики (appraisers) refers to people, the collective form is a very natural choice here.
How do collective numerals like четверо decline in oblique cases?
Once you leave the Nominative and Accusative cases, collective numerals decline exactly like plural adjectives. To form the Instrumental case for четверо, you just drop the and add the standard plural adjective ending -ыми, giving you четверыми.
In the Nominative case, we would say четверо оценщиков (using the Genitive plural). Why isn't the noun in the Genitive case here?
This is a vital rule for Russian numerals: the rule where numerals force the following noun into the Genitive case only applies in the Nominative and inanimate Accusative. In all other (oblique) cases, the numeral stops controlling the noun, and instead, they both simply agree with the required case. Because the preposition с requires the Instrumental case, both the numeral (четверыми) and the noun (оценщиками) take the Instrumental.
I recognize the suffix -щик in оценщиками. What is the root of this word?
You correctly spotted the profession suffix -щик, which works just like the one in уборщик (cleaner) or переводчик (translator). The base comes from the noun оценка (evaluation or appraisal), which in turn comes from цена (price). Therefore, an оценщик is someone who determines the price or value of something.

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