3. Use the pigeonhole principle to answer the following questions. (a) The population of India is 1.36 billion. Show that there must be a set of people of size least 128400, all of whom have the same birthday and live in the same state. (There are 29 states in India.) (b) As humans colonize the galaxy, the number of habitated planets grows as the recurrence: P(t +1) 1.1P(t) +7, where t denotes years after the start of colonization. The human population growth asymptotically behaves like O(2). Show that some planet must have a human population of at least (1.8t). (c) Dice can have the following properties: number of sides (at least 6 and at most 24), color (red, blue, green, or yellow), and country of manufacture (China, USA, or Japan What is thc largest set of die that you could possibly have such that no 50 of them share all three attributes in common? 3. Use the pigeonhole principle to answer the following questions. (a) The population of India is 1.36 billion. Show that there must be a set of people of size least 128400, all of whom have the same birthday and live in the same state. (There are 29 states in India.) (b) As humans colonize the galaxy, the number of habitated planets grows as the recurrence: P(t +1) 1.1P(t) +7, where t denotes years after the start of colonization. The human population growth asymptotically behaves like O(2). Show that some planet must have a human population of at least (1.8t). (c) Dice can have the following properties: number of sides (at least 6 and at most 24), color (red, blue, green, or yellow), and country of manufacture (China, USA, or Japan What is thc largest set of die that you could possibly have such that no 50 of them share all three attributes in common