The basic formulas are:

Available RAM = Global Buffers + (Thread Buffers x max_connections)
max_connections = (Available RAM – Global Buffers) / Thread Buffers

To get the list of buffers and their values:

SHOW VARIABLES LIKE '%buffer%';

Here’s a list of the buffers and whether they’re Global or Thread:

Global Buffers: key_buffer_size, innodb_buffer_pool_size, innodb_log_buffer_size, innodb_additional_mem_pool_size, net_buffer_size, query_cache_size
Thread Buffers: sort_buffer_size, myisam_sort_buffer_size, read_buffer_size, join_buffer_size, read_rnd_buffer_size, thread_stack

MariaDB [(none)]> SHOW VARIABLES LIKE '%buffer%';
+---------------------------------------+-----------+
| Variable_name                         | Value     |
+---------------------------------------+-----------+
| aria_pagecache_buffer_size            | 134217728 |
| aria_sort_buffer_size                 | 134217728 |
| bulk_insert_buffer_size               | 8388608   |
| innodb_blocking_buffer_pool_restore   | OFF       |
| innodb_buffer_pool_instances          | 1         |
| innodb_buffer_pool_populate           | OFF       |
| innodb_buffer_pool_restore_at_startup | 0         |
| innodb_buffer_pool_shm_checksum       | ON        |
| innodb_buffer_pool_shm_key            | 0         |
| innodb_buffer_pool_size               | 134217728 |
| innodb_change_buffering               | all       |
| innodb_log_buffer_size                | 8388608   |
| join_buffer_size                      | 131072    |
| join_buffer_space_limit               | 2097152   |
| key_buffer_size                       | 16777216  |
| mrr_buffer_size                       | 262144    |
| myisam_sort_buffer_size               | 8388608   |
| net_buffer_length                     | 8192      |
| preload_buffer_size                   | 32768     |
| read_buffer_size                      | 262144    |
| read_rnd_buffer_size                  | 524288    |
| sort_buffer_size                      | 524288    |
| sql_buffer_result                     | OFF       |
+---------------------------------------+-----------+

Lets find out the RAM:

# free -b
              total        used        free      shared  buff/cache   available
Mem:     3975184384   978608128  1691045888     9445376  1305530368  2661937152

Lets get our data together.

RAM = 3975184384
Global Buffers: key_buffer_size, innodb_buffer_pool_size, innodb_log_buffer_size, innodb_additional_mem_pool_size, net_buffer_length, query_cache_size
or, from above…
Global Buffers: 16777216 + 134217728 + 8388608 + 0 + 8192 + 0 = 159391744

Thread Buffers: sort_buffer_size, myisam_sort_buffer_size, read_buffer_size, join_buffer_size, read_rnd_buffer_size, thread_stack
or, from above…
Thread Buffers: 524288 + 8388608 + 262144 + 131072 + 524288 + 0 = 9830400

With this information, the following is the calculation:
max_connections = (Available RAM – Global Buffers) / Thread Buffers
max_connections = (3975184384 – 159391744) / 9830400

So the formula shows 378 Max Connections on this machine

Test info with mysqltuner

Log into your server with a root or sudo user via SSH.
Download MySQLTuner by executing the following command:

wget -O mysqltuner.pl https://raw.githubusercontent.com/major/MySQLTuner-perl/master/mysqltuner.pl --no-check-certificate

Give the script 775 permissions:

chmod 775 mysqltuner.pl

Run the script with the following command:

perl mysqltuner.pl

Resources:
Handy Calculator Download: https://journeyontux.wordpress.com/2011/12/22/calculate-number-of-connections-for-mysql-server/
Another Calculator: http://www.mysqlcalculator.com/