Count post with tags within one category?

Yes it is possible to count the posts with certain tags within a particular category. You can achieve this by using a combination of wp functions like WP_Query to retrieve posts and get_terms to retrieve the tags within the specified category.

<?php
$category_slug = 'donotallowpost';
$term_slugs = [ 'cat', 'bird', 'chinchilla', 'dog', 'ferret', 'hamster', 'horse', 'lizard', 'mouse', 'pig', 'poultry', 'rabbit', 'snake', 'turtle' ];

// Get the category ID using the slug
$category = get_category_by_slug($category_slug);
$category_id = $category->term_id;

// Get the tags within the specified category
$tags = get_terms( [
    'taxonomy' => 'post_tag',
    'hide_empty' => false, // Make sure to include tags with no posts
    'object_ids' => get_objects_in_term( $category_id, 'category' ), // Get tags associated with the category
] );

// Initialize post count
$post_count = 0;

// Loop through each tag to count posts
foreach ($tags as $tag) {
    $tag_post_count = new WP_Query( array(
        'tag_id' => $tag->term_id,
        'posts_per_page' => -1, // Retrieve all posts
        'fields' => 'ids', // Only retrieve post IDs for performance
    ) );
    $post_count += $tag_post_count->post_count;
}

echo $post_count;
?>