Please try the following solution.

Window function ROW_NUMBER() is very handy for such scenario.

SQL

-- DDL and sample data population, start
DECLARE @tbl TABLE (Category VARCHAR(20),Serial_Number VARCHAR(10));
INSERT @tbl (Category, Serial_Number) VALUES
('Sports', '3462'),
('Sports', '4373'),
('Sports', '3453'),
('Auto', '7742'),
('Auto', '1247'),
('Auto', '3463'),
('Auto', '1242'),
('Auto', '7746')
-- DDL and sample data population, end

;WITH rs AS
(
    SELECT *
        , ROW_NUMBER() OVER (PARTITION BY Category ORDER BY @@spid) AS seq
        , COUNT(*) OVER (PARTITION BY Category ORDER BY @@spid) AS cnt
    FROM @tbl
)
SELECT Category, Serial_Number, cnt
FROM rs
WHERE seq = 1;

Output

The following might work if you were to return a random row in a group

CREATE TABLE abcDB (
    Category VARCHAR(20),
    Serial_Number INT
);

INSERT INTO abcDB (Category, Serial_Number)
VALUES
    ('Sports', 3462),
    ('Sports', 4373),
    ('Sports', 3453),
    ('Auto', 7742),
    ('Auto', 1247),
    ('Auto', 3463),
    ('Auto', 1242),
    ('Auto', 7746);

WITH main_table AS (
  SELECT
    Category,
    Serial_Number AS Example_Serial_Number,
    COUNT(*) OVER (PARTITION BY Category) AS Serial_Number_Count,
    DENSE_RANK() OVER (PARTITION BY Category ORDER BY Serial_Number) AS ranking
  FROM abcDB
)

SELECT
  Category,
  Serial_Number_Count,
  Example_Serial_Number
FROM main_table
WHERE ranking = FLOOR(RAND()*(Serial_Number_Count-1+1)+1);

fiddle